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WHITE'S    GRADED-8CHOOL    SE7iJjS3'^ 


COISIPLETE 


AEITHMETIC, 


MENTAL  AND  WRITTEN  EXERCISES 


NATURAL  SYSTEM  OF  INSTRUCTION. 


By  E.  E.  AVHITE,  M.  A. 


CINCINNATI: 

WILSON,    H  INKLE   &    CO 

NEW  YORK:  CLARK  &  MAYNARD. 


Entered  according  to  Act  of  Congress,  in  the  year  1870,  by 

WILSON,  HINKLE  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for 

the  Southern  District  of  Ohio. 


aPUCATIOS^^  OEFT, 


EtECTnOTVPED  AT 

THE  FRA.NKMN  TYPE  FOUN'DUY, 

CINCINNATI. 


CUL 


PREFACE. 


This  work  is  called  a  Complete  Arithmetic,  because  it  em- 
braces all  the  subjects  which  properly  belong  to  a  school  arith- 
metic, and  because  it  treats  these  subjects  both  analytically  and 
inductively.  It  is  designed  to  be  a  complete  text-book  for  pupils 
who  have  mastered  the  elements  of  numbers. 

The  work  is  characterized  by  the  same  features  as  the  lower 
books  of  the  series,  viz. : 

1.  It  combines  3Iental  and  Written  Arithmetic  in  a  practical  and 
philosophical  manner.  This  is  done  by  making  the  mental  exer- 
cises preparatory  to  the  written ;  and  thus  these  two  classes  of 
exercises,  which  have  been  so  long  and  so  unnaturally  divorced, 
are  united  as  the  essential  complements  of  each  other. 

2.  It  faithfally  embodies  the  inductive  method  of  teachinff.  The 
written  methods  are  preceded  by  the  analysis  of  mental  problems, 
and  both  the  written  methods  and  the  principles  which  they  in- 
volve, are  derived  inductively  from  the  analytic  processes.  The 
successive  steps  of  each  process  are  mastered  by  the  pupil 
through  the  solution  of  problems,  and  he  is  required  to  deduce 
and  state  the  rules  before  he  is  confronted  with  the  author's 
generalization.  All  definitions  which  are  deducible  from  the 
processes,  and,  with  few  exceptions,  all  principles  and  rules,  are 
placed  after  the  j^roblems— a  feature  peculiar  to  this  Series. 

3.  It  is  specially  adapted,  both  in  matter  and  method,  to  the  grade  of 
pupils  for  ivhich  it  is  designed.  The  greater  portion  of  the  work 
is  devoted  to  a  progressive  and  thorough  treatment  of  subjects 
not  embraced  in  the  lower  books  —  an  arrangement  which  spe- 

54t>{49  (iii) 


IV    •    '      '  '  PREFACE. 

^ciall'y  mte6fe'  the  ivataits  of  Graded  Schools.  But  sixty  pages  cover 
the  same  ground  as  the  Intermediate  Arithmetic,  and  of  these 
not  more  than  twenty  pages  are  in  any  sense  a  repetition.  The 
repeated  matter  consists  of  definitions,  principles,  and  rules,  all 
the  problems  being  new.  The  subjects  before  treated  are  not 
only  concisely  reviewed,  but  from  a  higher  stand-point.  Of  the 
twenty-four  pages  devoted  to  the  fundamental  rules,  eight  present 
new  abbreviated  methods;  and  of  twenty-eight  pages  devoted  to 
Denominate  Numbers,  simple  and  compound,  more  than  sixteen 
discuss  new  topics.  A  similar  difference  is  observable  in  the  treat- 
ment of  Common  Fractions  and  United  States  Money.  Among 
the  added  articles  worthy  of  special  mention  are  those  on  Denom- 
inate Fractions,  the  Metric  System,  and  Longitude  and  Time. 

In  the  number  of  problems,  the  author  has  aimed  to  hit  the 
golden  mean  between  a  paucity  and  an  excess,  and  the  greatest 
pains  have  been  taken  to  make  them  sufiiciently  progressive, 
varied,  and  difficult,  to  afford  the  requisite  drill  and  practice. 
Instead  of  rehashing  old  problems,  with  their  incorrect  data  and 
obsolete  terms,  the  author  has  gone  to  science  and  history  for 
statistical  information  of  practical  value,  and  he  has  aimed  to 
present  the  current  values,  terms,  forms,  and  usages  of  American 
business.  The  mental  problems  will  be  found  as  difficult  and 
comprehensive  as  those  which  constitute  the  latter  half  of  the 
standard  Mental  Arithmetics,  and  are  sufficiently  numerous  to 
afford  thorough  drills  in  analysis. 

The  explanations  of  the  written  processes  are  not  designed  to 
serve  as  models  for  the  pupil  to  memorize  and  repeat.  They  are 
intended  to  supplement  the  analysis.  In  some  cases,  a  formal 
analysis  is  given ;  in  others,  a  principle  is  deduced  or  demon- 
strated; and  in  others,  the  process  is  described  or  its  principles 
stated.  Neither  teacher  nor  pupil  is  denied  the  privilege  of  de- 
termining his  own  explanations. 

Another  characteristic  feature  of  this  work  is  the  prominence 


PREFACE.  V 

given  to  Principles.  A  clear  comprehension  of  the  principles 
of  arithmetic  is  essential  to  its  thorough  mastery,  and  their  in- 
duction, proof,  and  illustration  are  mental  exercises  of  great 
value.  Until  the  pupil  can  step  inductively  from  processes  to 
principles,  he  has  not  a  thorough  knowledge  of  numbers.  In  this 
work  the  principles  are  concisely  and  formally  stated  in  connec- 
tion with  the  rules  which  are  based  upon  them. 

The  author  invites  special  attention  to  the  sections  treating 
of  Percentage,  Ratio  and  Proportion,  and  Involution  and  Evo- 
lution. Over  eighty  pages  are  devoted  to  Percentage  and  its 
applications,  and  it  is  believed  that  the  treatment  will  be  found 
not  only  full  and  thorough,  but  of  great  practical  value.  The 
student  who  masters  these  pages  will  certainly  have  a  fair 
knowledge  of  the  nature,  laws,  and  usages  of  the  business  of  the 
country.  The  introduction  of  Formulas,  it  is  hoped,  Avill  prove  a 
useful  feature. 

The  thorough  treatment  of  Ratio  before  Proportion,  and  of  the 
latter  before  its  application  to  the  solution  of  problems,  will  make 
the  mastery  of  this  subject  easy.  The  treatment  of  Involution 
and  Evolution  will  not  escape  notice.  The  geometrical  explana- 
tions of  Square  Root  and  Cube  Root  are  the  reverse  of  those 
usually  given,  and  are  believed  to  be  new.  They  will  be  found 
both  simple  and  natural. 

All  useless  and  obsolete  subjects  have  been  omitted;  and  those 
of  interest  only  to  more  advanced  students  and  teachers,  are 
presented  in  an  appendix.  The  typography  and  illustrations  are 
beautiful  and  appropriate. 

The  Complete  Arithmetic  is  submitted  to  American  teach- 
ers in  the  hope  that  it  may  not  only  be  found  new  in  its  general 
plans  and  in  many  of  its  methods  and  details,  but  that  it  may 
prove  eminently  adapted  to  the  present  wants  and  condition  of 
Graded  Systems  of  Instruction. 

Columbus,  Ohio,  July,  1870. 


SUGGESTIONS  TO  TEACHERS. 

1.  The  Mental  Problems  should  be  made  a  thorough  drill  in  analysis; 
but,  since  the  reasoning  faculty  is  not  trained  by  mere  logical  verbiage, 
the  solution  should  be  concise  and  simple.  They  should  also  be  made 
introductory  to  the  written  processes  of  which  they  are  often  a  com- 
plete elucidation.  The  corresponding  problems  in  the  two  classes  of 
exercises  should  be  recited  in  connection,  as  well  as  separately. 

2.  All  Written  Problems  should  be  solved  by  the  pupils  on  slate  or 
paper,  and  the  solutions  should  be  brought  to  the  recitation  for  the 
teacher's  inspection  and  criticism.  From  three  to  five  minutes  at  the 
beginning  of  the  recitation  will  suffice  to  compare  the  answers  of  the 
class,  and  ascertain  the  accuracy  and  neatness  of  each  pupil's  work. 
Time  thus  taken  from  the  class-drill  is  more  than  made  good  by  the 
increased  interest,  self-reliance,  and  accuracy,  which  the  absence  of 
answers  secures.  The  explanations  of  the  written  processes,  given 
by  the  pupil,  should  be  both  analytic  and  inductive. 

3.  The  Definitions  should  be  deduced  and  stated  by  the  pupils 
under  the  guidance  of  the  teacher,  and  this  can  usually  be  done  in 
connection  with  the  solution  of  the  problems.  See  Int.  Arith.,  p.  5, 
Sug.  3.  When  the  definitions  are  placed  before  the  problems,  as  in 
the  applications  of  Percentage,  they  should  be  studied  by  the  pupils, 
but  their  recitation  may  be  deferred  until  the  problems  are  solved, 
and  the  processes  mastered. 

4.  The  Principles  should  be  taught  inductively,  when  this  is  pos- 
sible, and  each  should  be  proven  or  illustrated,  or  both,  by  the  pupil. 
A  thorough  mastery  of  every  principle  should  be  made  an  essential 
condition  of  the  pupil's  progress.  The  recitation  should  secure  a 
constant  application  of  known  principles,  and  a  clear  comprehension 
of  all  new  ones. 

5.  The  Rules  should  also  be  deduced  and  stated  by  the  pupils. 
The  true  order  is  this:  1.  A  mastery  of  the  process.  2.  Recognition 
of  the  successive  steps  in  order,  and  a  statement  of  each.  3.  Combi- 
nation of  these  several  statements  into  a  general  statement.  4.  Com- 
parison of  this  generalization  with  the  author's  rule.  5.  Memorizing 
of  the  rule  approved.     See  Int.  Arith.,  p.  6. 

6.  When  two  or  more  methods  or  solutions  are  given,  the  one  pre- 
ferred should  be  thoroughly  taught.  It  is  well  for  pupils  to  understantl 
different  processes  and  explanations,  but  they  should  be  made  familiar 
with  one  of  them. 

7.  Before  a  subject  is  left,  the  pupils  should  be  required  to  make 
a  topical  analysis  of  the  definitions,  principles,  and  rules,  and  the 
same  should  be  recited  with  accuracy  and  dispatch.  Their  knowl- 
edge of  the  subject  should  finally  be  tested  by  a  series  of  questions 
and  problems.     See  General  Review,  pp.  268-284. 

(vi) 


CONTENTS. 


SECTION  I -VI.— The  Fundamental  Rules. 


Notation  and  Numeration 
Addition     .... 
The  Addition  of  Two  Columns 
Subtraction  .         . 


PAGE 

10 
13 
15 
17 


Multiplication     . 
Abbreviated  Processes 
Division 
Abbi-cviated  Processes 


PAGE 

.  20 
.  22 
.     25 

.     28 


SECTION  VII.— Properties  of  Numbers. 


Divisors    and  Factors 
Cancellation 


32    Greatest  Common  Divisor  .         .     37 
35  I  Least  Common  Multiple  •  .         .40 


Notation  and  Numeration  .  43 

Reduction  of  Fractions       .  .  46 

Addition  of  Fractions        .  .  53 

Subtraction  of  Fractions    .  .  55 

Multiplication  of  Fractions  .  57 


SECTION  VIII.— Fractions. 

Division  of  Fractions 
Complex  Fractions 


Numbers  Parts  of  Other  Num- 
bers   66 

Review  of  Fractions  .         .     68 


SECTION  IX.— Decimal  Fractions. 


Numeration  and  Notation  .     73 

Reduction  of  Decimals        .         .     79 
Addition  of  Decimals         .         .     82 


Subtraction  of  Decimals  .  .  83 
Multiplication  of  Decimals  .  84 
Division  of  Decimals  .         .     85 


SECTION    X.- 

Notation  and  Reduction  . 
Addition  and  Subtraction  . 
Multiplication  and  Division 


■United  States  Money. 

Abbreviated  Methods 
Aliquot  Parts 


Bill. 


Surfaces 


SECTION    XI.— Mensuration. 
.  100  I  Solids  . 


104 


SECTION    XII.— Denominate  Numbers. 


Reduction 107 

Denominate  Integers  and  Mixed 
Numbers  .         .         .         .107 


Denominate  Fractioi 
Tlio  Metric  System 
Metric  Tables 


110 
119 
120 


(Vii) 


Vlll 


CONTENTS. 


SECTION    XIII.— Compound  Numbers. 


Addition  and  Subtraction 
Multiplication  and  Division 


PAGE 

125 
128 


Longitude  and  Time 


SECTION   XIV.— Percentage. 


The  Four  Cases  of  Percentage  137 

Review  of  the  Four  Cases       .  144 

Applications  of  Percentage      .  146 

Profit  and  Loss           .         .  146 

Commission  and  Brokerage  149 

Capital  and  Stock      .         .  154 

Insurance    ....  159 

Life  Insurance    .         .         .  163 

Taxes  .         .         .         .164 

Customs  or  Duties     .         .  168 

Bankruptcy         .         .         .  170 

Interest      .         .         .         .         .  171 

General  Method          .         .  172 


Six  Per  Cent.  Method 
Method  by  Days 
Partial  Payments 
The  Problems  in  Interest 
Review  of  Problems  . 

Present  Worth  and  Discount 

Bank  Discount 

Promissory  Notes  and  Drafts 

Bonds         .... 

Annual  Interest 

Compound  Interest   . 

Equation  of  Payments     . 

Equation  of  Accounts 


SECTION   XV.— Ratio  and  Proportion. 


Ratio  .         .         .         ,         .  220 

Proportion  ....  224 

Simple  Proportion     .         .         .  225 

Compound  Proportion       .         .  230 


Partnership 
Simple  Partnership  . 
Compound  Partnership 
Problems  for  Analysis 


SECTION   XVI. — Involution  and  Evolution. 

Involution  ....  246  i  Geometrical  Explanation 

Another  Method  of  Involution  248    Cube  Root 

Evolution  ....  249  '  Geometrical  Explanation 

Square  Root      ....  251  I  Applications 

SECTION   XVII.— General  Review. 
Test  Problems   .         .         .         .     268  [  Test  Questions  . 

APPENDIX. 


Notation 287 

Proofs  by  Excess  of  9's     .         .  287 

Circulating  Decimals         .         .  289 

Tables  of  Denominate  Numbers  291 

Legal  Rates  of  Intcrrst    .         .  295 

Life  Insurance  .         .         .  295 

Equation  of  Payments      .         .  296 

Arithmetical  Progression  .  297 

Geometrical  Progression  .         .  300 


Alligation 

Duodecimals 

Permutations     . 

Annuities  . 

Rules  of  Mensuration 

Henkle's    Method    of    Writing 

Decimals         .... 
Schuyler's   Contracted   Method 

of  Multiplying  Decimals 


page 
130 


175 
179 
181 
185 
190 
192 
194 
198 
204 
205 
208 
211 
215 


234 
235 
237 
239 


255 
257 
262 
264 


281 


303 
306 
308 
309 
309 

311 

312 


COMPLETE  ARITHMETIC. 


SECTION    I. 

PRELIMINARY  DEFINITIONS. 

Art.  1.  Arithmetic  is  the  science  of  numbers,  and  the 
art  of  numerical  computation. 

As  a  science,  Arithmetic  treats  of  the  relations,  properties,  and  prin- 
ciples of  numbers ;  and,  as  an  art,  it  applies  the  science  of  numbers 
to  their  computation. 

2.  A  Unit  is  one  thing  of  any  kind. 

3.  A  dumber  is  a  unit  or  a  collection  of  units. 

4.  An  Iwtegev  is  a  number  composed  of  whole  or  integral 
units ;  as,  5,  12,  20.     It  is  also  called  a  Whole  Number. 

5.  Numbers  are  either  Concrete  or  Abstract 

A  Concrete  I^timher  is  applied  to  a  particular  thing 
or  quantity ;   as,  4  stars,  6  hours. 

When  a  concrete  number  expresses  the  denominate  units  of  cur- 
rency, weiglit,  or  measure,  it  is  called  a  Denominate  Number.  (Art.  174.) 

An  Abstract  Wumber  is  not  applied  to  a  particular 
thing  or  quantity ;  as,  4,  6,  20. 

A  concrete  number  is  composed  of  concrete  units,  and  an  abstract 
number  of  abstract  units. 

(9) 


10  COMPLETE   ARITHMETIC. 

;;R.  A  JProMem^is  a  question  proposed  for  solution. 

7.  An  lExanipl6  is  a  problem  used  to  illustrate  a  process 
or.a.principi»^j     ;  ;- 

8.  A  Itiile  is  a  general  description  of  a  process. 

9.  An  Arithmetical  Sign  is  a  character  denoting  an 
operation  to  be  performed  upon  numbers,  or  a  relation 
between  them. 

10.  In  the  Mental  Solution  of  a  problem,  the  suc- 
cessive steps  are  determined  mentally,  and  the  results  are 
held  in  the  mind. 

In  the  Written  Solution  of  a  problem,  the  results  are 
written  on  a  slate,  paper,  or  other  substance. 


SECTION    II. 
NOTATION  AND   NUMERATION. 

MENTAL    EXERCISES. 

1.  How  many  hundreds,  tens,  and  units  in  368?  427? 
549?  608?  724?  806?  870? 

2.  How  many  hundred-thousands,  ten-thousands,  and 
thousands  in  456048?  607803?  680435?  700450? 

3.  Read  the  thousands'  period  in  3045;  40607;  150482; 
405360;  920400;  600060. 

4.  Read  first  the  thousands'  period  and  then  the  units' 
period  in  65671 ;  120408  ;  400750  ;  650400 ;  80008. 

5.  Read  45037406;  520600480;  138405050. 

6.  Read  50008140 ;  600650508;  805000030. 

7.  Read  5308008450;  35006060600;  120500408080. 

8.  Read  7008360004;  302000860060;  500080800008. 


NUMERATION    AND   NOTATION.  11 

WRITTEN    EXERCISES. 

9.  Express  in  figures  the  number  composed  of  5  thou- 
sands, 7  tens,  and  3  units;  4  ten-thousands,  6  hundreds, 
and  5  units. 

10.  Express  in  figures  50  thousands  and  40  units;  406 
thousands  and  30  units;   700  thousands  and  7  units. 

Express  the  folloAving  numbers  in  figures: 

11.  Five  million  five  thousand  five  hundred. 

12.  Sixty  million  sixty  thousand  and  sixty. 

13.  Seven  hundred  million  seven  hundred  thousand  seven 
hundred. 

14.  Five  hundred  and  sixty  million  sixty-eight  thousand. 

15.  Four  billion  fi)urteen  million  fi)rty-five  thousand. 

16.  Sixty-five  billion  six  thousand  and  fifty. 

17.  Three  hundred  and  fifty  billion  forty-nine  million. 

18.  Seventeen  trillion  seventy  billion  seven  hundred 
thousand  four  hundred. 

19.  Fifty-six  trillion  sixteen  million  and  ninety. 

20.  Seven  quadrillion  eighty-five  billion  two  hundred  and 
four. 

DEFINITIONS  AND  TRINCIPLES. 

11.  There  are  three  methods  of  expressing  numbers : 

1.  By  words;  as,  five,  fifty,  etc. 

2.  By  letters,  called  the  Roman  method. 

3.  By  figures,  called  the  Arabic  method. 

12.  Notation  is  the  art  of  expressing  numbers  by  fig- 
ures or  letters. 

13.  Nmneration  is  the  art  of  reading  numbers  ex- 
pressed by  figures  or  letters. 

Note. — Notation  may  be  defined  to  be  the  art  of  writing  numbers, 
and  Numeration,  the  art  oi  reading  numbers.  In  Arithmetic,  the  term 
notation  is  used  to  denote  tlie  Arabic  method. 

14.  In  the  Roman  Notation,  numbers  are  expressed  by 
means  of  seven  capital  letters,  viz:  I,  V,  X,  L,  C,  D,  M. 


12  COMPLETE   ARITHMETIC. 

I  stands  for  one;  V  for  five;  X  for  ten;  L  for  fifty;  C 
for  one  hundred;  D  for  five  hundred;  M  for  one  thousand, 
All  other  numbers  are  expressed  by  repeating  or  combining 
these  letters. 

15.  In  the  Arabic  Notation,  numbers  are  expressed  by 
means  of  ten  characters,  called  figures,  viz:  0,  1,  2,  3,  4, 
5,  6,  7,  8,  9. 

The  first  of  these  characters,  0,  is  called  Naught,  or  Cipher. 
It  denotes  nothing,  or  the  absence  of  number. 

The  other  nine  characters  are  called  Significant  Figures, 
or  Numeral  Figures.  They  each  express  one  or  more  units. 
They  are  also  called  Digits. 

16.  The  successive  figures  which  express  a  number,  denote 
successive  Orders  of  Units.  A  figure  in  units'  place  denotes 
units  of  the  first  order;  in  tens'  place,  units  of  the  second  order; 
in  hundreds'  place,  units  of  the  third  order,  and  so  on — the 
term  units  being  used  to  express  ones  of  any  order. 

17.  Figures  have  two  values,  called  Simple  and  Local. 
The  Simple  Value  of  a  figure  is  its  value  when  stand- 
ing alone.     It  is  also  called  its  Absolute  value. 

The  Local  Value  of  a  figure  is  its  value  arising  from 
the  order  in  which  it  stands.  The  local  value  of  a  figure  is 
tenfold  greater  in  hundreds'  order  than  in  tens'  order. 

18.  The  local  value  of  each  of  the  successive  figures  Avhich 
express  a  number,  is  called  a  Term.  The  terms  of  325  are 
3  hundreds,  2  tens,  and  5  units. 

19.  The  figures  denoting  the  successive  orders  of  units, 
are  divided  into  groups  of  three  figures  each,  called  Periods. 
The  first  or  right-hand  period  is  called  Units;  the  second. 
Thousands;  the  third.  Millions;  the  fourth.  Billions;  the  fifth, 
Trillions;  the  sixth.  Quadrillions;  the  seventh,  Quintillions ; 
the  eighth,  Sextillions;  the  ninth,  Septillions;  the  tenth.  Oc- 
tillions; the  eleventh,  Nonillions ;  the  twelfth,  Decillions,  etc. 

Note. — The  division  of  orders  into  periods  of  three  figures  each  is 
the  French  method.     In  the  English  method,  the  period  contains  six 


ADDITION.  13 

orders,  the  name  of  the  first  period  being  Units,  the  second  Millions; 
the  third  Billions,  etc. 

20.  The  three  orders  of  any  period,  counting  from  the 
right,  denote  respectively  Units,  Tens,  and  Hundreds  of  that 
period.  They  may  be  briefly  read  by  calling  the  first  order 
by  the  name  of  the  period,  and  uniting  the  words  ten  and 
hundred  in  each  period  after  the  first  with  the  period's  name. 

Thus,  the  orders  of  thousands'  period  may  be  read  thousands, 
ten-thousands y  hundred-thousands;  the  orders  of  millions'  pe- 
riod, millions,  ten-millions,  hundred-millions,  etc. 

21.  Principles. — 1.  Ten  units  of  the  first  order  make  one 
unit  of  the  second  order,  ten  units  of  the  second  order  make 
one  unit  of  the  third,  and,  generally,  ten  units  of  any  order 
make  one  unit  of  the  next  higher  order.      Hence, 

2.  The  value  of  the  successive  orders  of  figures  increases  ten- 
fold from  right  to  left. 

3.  The  value  of  a  figure  is  increased  tenfold  by  each  removal 
of  it  one  order  to  the  left,  and  is  decreased  tenfold  by  each  re- 
moval of  it  one  order  to  the  right. 


SECTION    III. 
ADDITION. 

MENTAL  PROBLEMS. 

1.  Add  by  6's  from  1  to  73,  thus:   1,  7,  13,  19,  25,  etc. 

2.  Add  by  7's  from  3  to  73;  from  6  to  90. 

3.  Add  by  8's  from  5  to  77;  from  7  to  95. 

4.  Add  by  9's  from  4  to  76;  from  8  to  98. 

5.  The  ages  of  five  boys  are  respectively  12,  10,  9,  8,  and 
7  years:  what  is  the  sum  of  their  ages? 

6.  A  rode  45  miles  the  first  day,  42  miles  the  second  day, 
and  38  miles  the  third  day :  how  far  did  he  ride  in  all  ? 

Suggestion. — Add  the  tens  and  then  the  units  of  each  couplet,  thus: 
45  H-  40  =  85,  85  +  2  -=  87 ;  87  +  30  =  117,  117  +  8  =  125.     Or  name    - 
only  results,  thus:  45,  85,  87;  117,  125.     (Art.  22.) 


14  COMPLETE    ARITHMETIC. 

7.  A  drover  bought  37  sheep  of  one  farmer,  44  sheep  of 
another,  48  sheep  of  another,  and  27  sheep  of  another:  how 
many  sheep  did  he  buy? 

8.  The  Senior  class  of  a  college  contains  27  students,  the 
Junior  class  34,  the  Sophomore  class  38,  and  the  Freshman 
class  46:  how  many  students  in  the  college? 

9.  A  grocer  sold  18  sacks  of  flour  on  Monday,  23  on 
Tuesday,  27  on  Wednesday,  24  on  Thursday,  35  on  Friday, 
and  37  on  Saturday:  how  many  sacks  did  he  sell  during  the 
week? 

10.  A  lady  paid  $36  for  a  carpet,  $34  for  a  bureau,  $16 
for  awashstand,  $28  for  a  bedstead,  and  $42  for  chairs:  how 
much  did  she  pay  for  all? 

11.  A  man  paid  $85  for  a  horse,  and  $17  for  his  keeping; 
and  then  sold  him  so  as  to  gain  $15:  for  how  much  did  he 
sell  the  horse? 

12.  Two  men  start  from  the  same  point,  and  travel  in 
opposite  directions,  the  one  at  the  rate  of  54  miles  a  day, 
and  the  other  at  the  rate  of  48  miles  a  day:  how  far  will 
they  be  apart  at  the  close  of  the  second  day? 

"WRITTEN   PROBLEMS. 

13.  Add  347,  4086,  7080,  29408,  and  67736. 

14.  667  +  3804  +  45608  -f  304867  +  87609  =  what? 

15.  Add  four  thousand  and  fifty-six ;  sixty-three  thousand 
seven  h\indred ;  seven  million  nine  thousand  and  ninety-nine ; 
and  fifty-six  million  nine  hundred  and  seventy-eight. 

16.  Add  eight  million  eighty  thousand  eight  hundred ; 
seven  hundred  thousand  and  seventy;  five  million  eighty-six 
thousand  seven  hundred  and  eight;  and  sixty  million  six 
hundred  thousand  and  seventy. 

17.  A  grain  dealer  bought  wheat  as  follows:  Monday, 
2480  bushels;  Tuesday,  788  bushels;  Wednesday,  1565 
bushels;  Thursday,  2684  bushels;  Friday,  985  bushels;  Sat- 
urday, 3867  bushels.  How  many  bushels  did  he  buy  during 
the  week? 


ADDITION.  15 

18.  Ohio  contains  39964  square  miles;  Indiana,  83809; 
Illinois,  55409 ;  Michigan,  56243 ;  Wisconsin,  53924 ;  Min- 
nesota, 83531 ;  Iowa,  55045  ;  and  Missouri,  65350.  What  is 
the  total  area  of  these  eight  States  ? 

19.  The  population  of  these  States  in  1860  was  as  follows : 
Ohio,  2339511 ;  Indiana,  1350428 ;  Illinois,  1711951 ;  Michi- 
gan, 756890  ;  Wisconsin,  778714  ;  Minnesota,  189923  ;  Iowa, 
674913;  Missouri,  1182012.  What  was  their  total  popu- 
lation ? 

20.  The  territory  of  the  United  States  has  been  acquired 
as  follows : 

Square  miles. 

Territory  ceded  by  England,  1783,           ....  815615 

Louisiana,  as  acquired  from  France,  1803,           .         .         .  930928 

Florida,  as  acquired  from  Spain,  1821,      ....  59268 

Texas,  as  admitted  to  the  Union,  1845,        ....  237504 

Oregon,  as  settled  by  treaty,  1846,             ....  280425 

California,  etc.,  as  conquered  from  Mexico,  1847,        .         .  049762 

Arizona,  as  acquired  from  Mexico  by  treaty,  1854,          .  27500 

Alaska,  as  acquired  from  Russia  by  purchase,  1867,            .  577390 

What  is  the  total  area  of  the  United  States  ? 

ADDITION  OF  TWO   COLUMNS. 

22.  There  is  a  practical  advantage  in  adding  two  columns 
at  one  operation.  Some  accountants  add  three  or  more  col- 
umns in  this  manner. 

21.  Add  67,  58,  43,  36,  and  54. 

Process. 

67  Add  thus  :  54  +  30  -=  84,  +  6  =  90  ;   90  +  40  =  130,  + 

58  3  =  133 ;    133  +  50  ==  183,  -f  8  -^  191 ;  191  +  60  =  251, 

43  +  7  =  258. 

36  Or  thus,  naming  only  results:    54,  84,  90;    130,  133,  183; 

_54  191 ;  251,  258. 
258 

Note. — The  process  consists  in  first  adding  the  tens  of  each  couplet, 
and  then  the  units.  If  preferred,  the  units  may  first  be  added,  and 
then  the  tens.  Suflicient  practice  will  enable  the  accountant  to  add 
two  columns  without  separating  the  numbers  into  tens  and  units. 


16  COMPLETE  ARITHMETIC. 

22.  Add  37,  40,  63,  84,  67,  22,  and  70. 

23.  Add  95,  46,  77,  66,  88,  63,  33,  and  44. 

24.  Add  67,  76,  45,  54,  38,  83,  27,  and  72. 

25.  Add  68,  86,  97,  79,  86,  68,78,  and  87. 

26.  Add  45,  60,  57,  86,  83,  76,  49,  58,  and  84. 

27.  Add  56,  75,  83,  96,  69,  73,  37,  38,  and  205. 

28.  Add  27,  72,  33,  38,  69,  96,  75,  57,  and  336. 

29.  Add  235,  88,  77,  66,  55,  44,  33,  22,  and  11. 

30.  Add  405,  56,  43,  47,  74,  36,  63,  75,  and  6Q. 

31.  Add  46,  67,  72,  38,  99,  87,  65,  74,  and  88. 

32.  Add  73,  86,  47,  56,  69,  65,  58,  33,  52,  and  94. 

DEFINITIONS  AND  PRINCIPLES. 

23.  Addition  is  the  process  of  finding  the  sum  of  two 
or  more  numbers. 

The  Sum  of  two  or  more  numbers  is  a  number  contain- 
ing as  many  units  as  all  of  them,  taken  together.  It  is  also 
called  the  Amount 

24.  The  Sign  of  Addition  is  a  short  vertical  line  bi- 
secting an  equal  horizontal  line,  thus  :   -|-.     It  is  called  plus. 

25.  The  Sign  of  Equality  is  two  short  horizontal 
parallel  lines,  thus:  =.  It  is  read  equals  or  is  equal  to. 
Thus,  7  -[-  8  =  15  is  read  7  plus  8  equals  15. 

26.  Lihe  Numbers  are  composed  of  units  of  the  same 
kind.  Thus,  4  balls  and  8  balls,  or  4  dimes  and  8  dimes,  or 
4  and  8,  are  like  numbers. 

27.  Principles. — 1.   Only  like  numbers  can  be  added. 

2.  Only  like  orders  of  figures  can  be  added. 

3.  Tlie  sum  is  of  the  same  hind  or  order  as  the  numbers  added. 

4.  The  sum  is  the  same  whatever  be  the  order  in  which  the 
numbers  are  added. 

Note. — See  appendix  for  method  of  proof  by  "casting  out  theO's." 


ADDITION.  17 

SECTION    IV. 
SUBTRACTION. 

MENTAL  PKOBLEMS. 

1.  Count  by  4's  from  61  back  to  1,  thus:  61,  57,  53,  etc. 

2.  Count  by  6's  from  53  back  to  5 ;  from  74  back  to  2. 

3.  Count  by  7's  from  66  back  to  3 ;  from  85  back  to  1. 

4.  Count  by  8's  from  75  back  to  3 ;  from  94  back  to  6. 

5.  Count  by  9's  from  73  back  to  1 ;  from  96  back  to  6. 

6.  A  grocer  having  a  certain  number  of  sacks  of  flour, 
bought  48  sacks,  and  sold  33  sacks,  and  then  had  34  sacks 
on  hand :  how  many  sacks  had  he  at  first  ? 

7.  A  man  sold  a  horse  for  $95,  which  was  $28  more  than 
the  horse  cost  him :  what  was  the  cost  of  the  horse  ? 

8.  Two  men  start  at  once  from  the  same  point,  and  travel 
in  the  same  direction,  one  traveling  52  miles  a  day,  and  the 
other  but  39  miles :  how  far  will  they  be  apart  at  the  close 
of  the  second  day? 

9.  A  man  earns  $85  a  month,  and  pays  $18  for  house 
rent,  and  $35  for  other  expenses :  how  much  does  he  save 
each  month? 

10.  A  gentleman  being  asked  his  age  said,  that  if  he 
should  live  27  years  longer,  he  should  then  be  three  score 
and  ten  :  what  was  his  age  ? 

11.  From  a  piece  of  carpeting  containing  68  yards,  a  mer- 
chant sold  27  yards  to  one  man  and  18  yards  to  another: 
how  many  yards  of  the  piece  were  left? 

12.  A  man  bought  a  carriage  for  $135,  paid  $21  for  re- 
pairing it,  and  then  sold  it  for  $170:   how  much  did  he  gain? 

13.  A  boy  earned  65  cents,  and  his  father  gave  him  33 
cents;  he  paid  45  cents  for  an  arithmetic  and  18  cents  for  a 
slate:  how  much  money  had  he  left? 

14.  There  are  85  sheep  in  three  fields;  there  are  36  sheep 

C.Ar.-2 


18  COMPLETE  ARITHMETIC. 

in  the  first  field,  and  28  sheep  in  the  second :  how  many  sheep 
in  the  third  field? 

15.  John  had  33  chestnuts,  and  Charles  25;  John  gave 
Charles  14  chestnuts,  and  Charles  gave  his  sister  as  many  as 
he  then  had  more  than  John:  how  many  chestnuts  did  the 
sister  receive? 

WRITTEN   PROBLEMS. 

16.  A  builder  contracted  to  build  a  school-house  for  $25460, 
and  the  job  cost  him  $21385:   what  were  his  profits? 

17.  The  earth's  mean  distance  from  the  sun  (old  value) 
is  95274000  miles,  and  that  of  Mars  is  145168136:  how 
much  farther  is  Mars  from  the  sun  than  the  earth? 

18.  The  population  of  Illinois  in  1860  was  1711951,  and 
in  1865  its  population  was  2141510 :  what  was  the  increase 
in  five  years? 

19.  The  population  of  Massachusetts  in  1860  was  1231066, 
and  in  1865  it  was  1267031 :  what  was  the  increase  in  five 
years  ? 

20.  The  area  of  the  Chinese  Empire  is  4695334  square 
miles,  and  the  area  of  the  United  States  is  3578392  square 
miles :  how  much  greater  is  the  Chinese  Empire  than  the 
United  States? 

21.  The  area  of  Europe  is  3781280  square  miles:  how 
much  greater  is  Europe  than  the  United  States?  The  Chi- 
nese Empire  than  Europe? 

22.  In  1866,  Ohio  produced  99766822  bushels  of  corn, 
and  Illinois  155844350  bushels :  how  many  bushels  did  Illi- 
nois produce  more  than  Ohio? 

23.  A  man  bought  a  farm  for  $5867,  and  built  upon  it  a 
house  at  a  cost  of  $1850,  and  then  sold  the  farm  for  $7250: 
how  much  did  he  lose? 

24.  An  estate  of  $13450  was  divided  between  a  widow  and 
two  children;  the  widow's  share  was  $6340,  the  son's  $1560 
less  than  the  widow's,  and  the  rest  fell  to  the  daughter: 
what  was  the  daughter's  share? 

25.  A  man  deposited  in  a  bank  at  one  time   $850,  at  an- 


SUBTRACTION.  19 

other,  $367,  and  at  another,  $670;  he  then  drew  out  $480, 
and  $375 :  how  much  money  had  he  still  in  bank  ? 

26.  A  man  bought  a  farm  for  $6450,  giving  in  exchange 
a  house  worth  $4500,  a  note  for  $1150,  and  paying  the  dif- 
ference in  money:  how  much  money  did  he  pay? 

27.  A  grain  dealer  bought  15640  bushels  of  wheat,  and 
sold  at  one  time  3465  bushels,  at  another,  4205,  and  at  an- 
other, 1080:  how  many  bushels  remained? 

28.  A  has  320  acres  of  land ;  B  has  Q5  acres  more  than 
A ;  C  has  124  acres  less  than  both  A  and  B  ;  and  D  has  as 
many  acres  as  both  A  and  C  less  the  number  of  acres  owned 
by  B.  How  many  acres  have  B,  C,  and  D  respectively? 
How  many  have  all? 

29.  From  the  sum  of  45003  and  13478,  take  their  differ- 
ence. 

DEFINITIONS  AND  PRINCIPLES. 

28.  Subtraction  is  the  process  of  finding  the  difference 
between  two  numbers. 

The  Difference  is  the  number  found  by  taking  one 
number  from  another.     It  is  also  called  the  Remainder. 

The  Winuend  is  the  number  diminished. 

The  Subtrahend  is  the  number  subtracted. 

29.  The  Sign  of  Subtraction  is  a  short  horizontal 
line,  made  thus  — .  It  is  called  minus.  Thus  12  —  5  is 
read  12  minus  5. 

30.  Principles. — 1.  The  7ninue7idj  subtrajiend,  and  differ- 
ence are  like  numbers. 

2.  The  minuend  is  the  sum  of  the  subtrahend  and  difference. 

3.  If  the  minuend  and  subtrahend  be  equally  increased,  the 
dfference  will  not  be  changed. 

4.  The  adding  of  10  to  a  term  of  the  minuend  and  1  to  the 
next  higher  term  of  the  subtrahend,  increases  the  minuend  and 
subtrahend  equally. 


20  COMPLETE  ARITHMET[C. 

SECTION      V. 

MULTIPLICATION. 

MENTAL    PROBLEMS. 

1.  There  are  24  hours  in  a  day :  how  many  hours  in  7 
days?     In  9  days?  11  days? 

2.  There  are  60  minutes  in  an  hour:  how  many  minutes 
in  8  hours?     In  12  hours?  15  hours? 

3.  If  a  man  earn  $63  a  month,  and  spend  848,  how  much 
will  he  save  in  12  months? 

4.  If  12  men  can  do  a  piece  of  work  in  15  days,  how  long 
will  it  take  one  man  to  do  it  ? 

5.  If  35  bushels  of  oats  will  feed  8  horses  25  days,  how 
long  will  they  feed  one  horse  ? 

6.  Two  men  start  from  the  same  place  and  travel  in  op- 
posite directions,  one  at  the  rate  of  28  miles  a  day,  and  the 
other  at  the  rate  of  32  miles  a  day :  how  far  will  they  be 
apart  at  the  end  of  five  days  ? 

7.  Two  men  are  450  miles  apart:  if  they  approach  each 
other,  one  traveling  30  miles  a  day  and  the  other  35  miles  a 
day,  how  far  apart  will  they  be  at  the  end  of  6  days  ? 

8.  A  cask  has  two  pipes,  one  discharging  into  it  90  gallons 
of  water  an  hour,  and  the  other  drawing  from  it  75  gallons 
an  hour:  how  many  gallons  of  water  will  there  be  in  the 
cask  at  the  end  of  12  hours? 

9.  A  had  $24,  B  four  times  as  much  as  A  less  $16,  and 
C  twice  as  much  as  A  and  B  together  plus  $17 :  how  much 
money  had  B  and  C? 

10.  A  farmer  sold  to  a  grocer  15  pounds  of  butter,  at  30 
cents  a  pound,  and  bought  8  pounds  of  sugar,  at  15  cents 
a  pound,  and  9  pounds  of  coffee,  at  20  cents  a  pound :  how 
much  was  still  due  him? 


MULTIPLICATION.  21 


WRITTEN  PROBLEMS. 


11.  Multiply  624  by  45  ;  by  405  ;  by  4005. 

12.  Multiply  38400  by  27  ;  by  607  ;  by  6007. 

13.  Multiply  7863  by  69  ;  by  6900 ;  by  64000. 

14.  Multiply  48000  by  760 ;  by  7600000. 

15.  There  are  5280  feet  in  a  mile :  how  many  feet  in  608 
miles  ?     In  3300  miles  ? 

16.  The  earth  moves  1092  miles  in  a  minute :  how  far 
does  it  move  in  1440  minutes,  or  one  day? 

17.  A  square  mile  contains  640  acres,  and  the  state  of 
Ohio  contains,  in  round  numbers,  40000  square  miles :  how 
many  acres  in  the  state  ? 

18.  If  a  garrison  of  380  soldiers  consume  56  barrels  of 
flour  in  75  days,  how  many  soldiers  will  the  same  amount 
of  flour  supply  one  day? 

19.  A  man  bought  a  farm,  containing  472  acres,  at  $24 
an  acre,  and  after  investing  $3450  in  buildings,  he  sold  the 
farm, at  $33  an  acre :  did  he  gain  or  lose,  and  how  much? 

DEFINITIONS  AND  PRINCIPLES. 

31.  JKultijilicatioil  is  the  process  of  taking  one  num- 
ber as  many  times  as  there  are  units  in  another. 

The  31ultlplicand  is  the  number  taken  or  multiplied. 

The  3Iultipliev  is  the  number  denoting  how  many 
times  the  multiplicand  is  taken. 

The  JProdlict  is  the  number  obtained  by  multiplying. 

The  multiplicand  and  multiplier  are  Factors  of  the  product,  and 
the  product  is  a  Multiple  of  each  of  its  factors. 

32.  The  Sif/n  of  Hiiltij^lication  is  an  oblique  cross, 
made  thus:    X-     It  is  read  multipled  hy,  or  times. 

When  placed  between  two  numbers,  it  shows  that  they  are  to  be 
multiplied  together ;  and,  since  the  order  of  the  factors  does  not  affect 
the  product,  either  number  may  be  made  the  multiplier.  The  mul- 
tiplier is  usually  written  after  the  sign. 


22  COMPLETE  ARITHMETIC. 

33.  The  product  may  be  obtained  by  adding  the  multipli- 
cand to  itself  as  many  times  less  one  as  there  are  units  in 
the  multiplier.  Hence,  Multiplication  is  a  short  7nethod  of 
finding  the  sum  of  several  equal  numbers. 

34.  Principles. — 1.  The  ^fultiplicand  may  be  either  con- 
crete or  abstract. 

2.  The  7mdtiplier  must  always  be  regarded  as  abstract. 

3.  The  product  and  multiplicand  are  like  numbers. 

4.  The  product  is  not  affected  by  changing  the  order  of  the 
factors. 

5.  The  midtiplicand  equals  the  product  divided  by  the  multi- 
plier. 

6.  The  midtiplier  equals  the  product  divided  by  the  midtipli- 
cand. 

7.  The  division  of  either  the  multiplicand  or  the  multiplier  by 
any  number  divides  the  product  by  that  number. 

ABBREVIATED  PROCESSES. 
Case  I. 

The    Mlialtiplier    lO,    lOO,    lOOO,    etc. 

1.  There  are  7  days  in  a  week:  how  many  days  in  10 
weeks?     In  100  weeks? 

2.  There  are  24  hours  in  a  day :  how  many  hours  in  10 
days?  100  days? 

3.  If  a  railway  train  run  30  miles  an  hour,  how  far  will 
it  run  in  10  hours?  1000  hours? 

4.  If  a  freight  car  will  carry  18  head  of  cattle,  how  many 
cattle  will  10  cars  carry?  100  cars?  1000  cars? 

5.  There  are  12  months  in  a  year:  how  many  months  in 
100  years?     1000  years? 

WKITTEN  PEOBLEMS. 

6.  Multiply  648  by  100. 

Process  :  648  X  100  —  64800.  The  annexing  of  a  cipher  to  a  num- 
ber removes  the  significant  figures  one  place  to  the  left,  and  hence 
increases  their  value  10  times ;  the  annexing  of  two  ciphers  removes 


MULTIPLICATION.  23 

the  significant  figures  two  places  to  the  left,  and  increases  their  value 
100  times.  Hence,  the  annexing  of  two  ciphers  to  648  multiplies  it 
by  100. 

7.  Multiply  456  by  10 ;  by  10000. 

8.  Multiply  3050  by  100 ;  100000. 

9.  Multiply  347000  by  1000  ;  by  1000000. 

10.  Multiply  889000  by  10000 ;  by  100. 

35.  Principle. — The  removal  of  a  figure  one  order  to  the 
left  increases  its  value  tenfold. 

36.  Rule.— To  multiply  by  10,  100,  1000,  etc.,  Annex  to 
the  multiplicand  as  many  ciphers  as  there  are  ciphers  in  the  mul- 
tiplier. 

Case  11. 

The    Mlultiplier    a   convenient   part   of   lO,    lOO, 
lOOO,   etc. 

Note. — If  the  class  is  not  sufficiently  familiar  with  the  subject  of 
fractions,  this  case  may  be  omitted. 

11.  There  are  24  sheets  of  paper  in  a  quire:  how  many 
sheets  in  2^  quires  ?     In  3^  quires  ? 

12.  There  are  60  minutes  in  an  hour :  how  many  minutes 
in  3i  hours?     In  12^  hours? 

13.  If  a  workman  earn  $40  a  month,  how  much  w411  he 
earn  in  2^  months?     In  12|^  months? 

14.  At  36  cents  a  yard,  what  will  25  yards  of  cloth  cost? 
33i  yards? 

15.  At  24  cents  a  dozen,  what  will  12|^  dozens  of  eggs 
cost?    16 J  dozens? 

WHITTEW  PEOBLEMS. 

16.  Multiply  459  by  33^. 

Process.  Since  33i  is  i  of  100,  33^  times  459  =  i  of  100 

3  )  45900  times  459  =  -^  of  45900.     Or,  multiply  the  multi- 

15300  Prod.         plicand  by  100,  and  divide  the  product  by  3. 

17.  Multiply  486  by  3^;  by  33^. 

18.  Multiply  1688  by  12^;  by  25;  by  50. 


24  COMPLETE  ARITHMETIC. 

19.  Multiply  40648  by  16|;  by  33i;  by  333^. 

20.  Multiply  3468  by  25  ;  by  125  ;  by  250. 

21.  Multiply  4086  by  16| ;  by  166|;  by  333|. 

22.  Multiply  10366  by  50 ;  by33|;  by  66^. 

37.  Principle. — Jj  the  multiplier  be  multiplied  by  a  given 
number,  and  the  resulting  product  be  divided  by  the  same  num- 
ber, the  quotient  ivill  be  the  true  product. 

38.  KuLE. — To  multiply  by  a  convenient  part  of  10,  100, 
1000,  etc.,  3Mti2}ly  by  10,  100,  1000,  etc.,  and  divide  the 
product  by  the  number  of  times  the  multiplier  has  been  increased. 

Case  III. 

The  MlTaltipliex-  a  little  less  tlxan  lO,  lOO,  lOOO,  etc. 

23.  Multiply  467  by  98. 

Process.  Since  98  =  100  —  2,  the  product  of  467  by  98  =  467 

46700  X  100  —  467  X  2,  or  46700  —  934.     In  multiplying 

^^4  \yy  100  the  multiplicand  is  taken  two  times  more  than 

45766,  Prod,  it  should  be. 

24.  Multiply  5672  by  99  ;  by  999. 

25.  Multiply  40863  by  97  ;  by  997. 

26.  Multiply  8679  by  998  ;  by  9998. 

27.  Multiply  618734  by  95 ;  by  99995. 

39.  Rule. — To  multiply  by  a  number  a  little  less  than  10, 
100,  1000,  etc.,  Multiply  by  10,  100,  1000,  etc.,  and  subtract 
from  the  product  tJie  multiplicand  midtiplied  by  the  difference  be- 
tween the  multiplier  and  10,  100,  1000,  etc.,  as  the  case  may  be. 

Case   IV. 

The  Mlxaltiplier  14,  15,  lO,  etc.,  or  31,  51,  61,  etc. 

28.  Multiply  7856  by  14 ;  by  41. 

1st  Process.  2d  Process. 

7856  X  14  7856  X  41 

31414  31414 


109974,  Product.  321996,  Pivduct. 

Note. — An  inspection  of  each  process  will  suggest  its  explanation. 
The  second  partial  product  need  not  be  written,  as  the  successive  terms 
can  be  added  mentally  to  the  proper  terms  of  the  first  partial  product. 


DIVISION.  25 

29.  Multiply  38407  by  13  ;  by  15  ;  by  17. 

30.  Multiply  4960  by  16 ;  by  18 ;  by  19. 

31.  Multiply  360978  by  31  ;  by  51 ;  by  71. 

32.  Multiply  48706  by  61 ;  by  81 ;  by  91. 

33.  Multiply  34087  by  17  ;  by  71 ;  by  18. 

40.  KuLES. — 1.  To  multiply  by  13,  14,  15,  etc.,  Multiply 
by  the  units'  term,  and  add  the  successive  products  after  the  first, 
ivhich  is  units,  to  the  successive  terms  of  the  midtiylicand. 

2.  To  multiply  by  31,  41,  51,  etc..  Multiply  by  the  tens'  tsrm, 
and  add  the  successive  products  to  the  successive  terms  of  the  mul- 
tiplicand beginning  with  tens. 


SECTION  VI. 

DIVISION. 

MENTAL   PROBLEMS. 


1.  There  are  7  days  in  a  week:  how  many  weeks  in  63 
days?  98  days?  126  days? 

2.  There  are  eight  quarts  in  a  peck:  how  many  pecks 
in  72  quarts?  120  quarts?  144  quarts? 

3.  There  are  60  minutes  in  an  hour :  how  many  hours  in 
480  minutes?  720  minutes?  1440  minutes? 

4.  A  man  paid  $3600  for  a  farm,  paying  at  the  rate  of 
$40  an  acre :   how  many  acres  in  the  farm  ? 

5.  A  grocer  bought  12  barrels  of  flour  for  $90,  and  sold 
them  so  as  to  gain  $18:  how  much  did  he  receive  per 
barrel  ? 

6.  Two  men  are  120  miles  apart,  and  are  traveling  toward 
each  other,  one  at  the  rate  of  7  miles  an  hour,  and  the 
other  at  the  rate  of  8  miles  an  hour :  in  how  many  hours 
will  they  meet? 

7.  If  a  man  can  build  a  wall  in  84  days,  how  long  will 
it  take  7  men  to  build  it? 

C.Ar.— ;^ 


26  COMPLETE  ARITHMETIC. 

8.  If  8  men  can  do  a  piece  of  work  in  15  days,  how  long 
will  it  take  12  men  to  do  it? 

9.  If  a  quantity  of  provisions  will  supply  a  ship's  crew 
of  20  men  15  weeks,  how  large  a  crew  will  it  supply  25 
weeks  ? 

10.  If  a  man  can  do  a  piece  of  work  in  40  days,  by  work- 
ing 8  hours  a  day,  how  long  would  it  take  him  if  he  should 
work  10  hours  a  day? 

11.  A  man  earns  $16  while  a  boy  earns  $9:  how  many 
dollars  will  the  man  earn  while  the  boy  is  earning  $12  ? 

12.  The  fore  wheels  of  a  carriage  are  each  9  feet  in  cir- 
cumference, and  the  hind  wheels  are  each  12  feet:  if  the 
fore  wheels  each  rotate  400  times  in  going  a  certain  distance, 
how  many  rotations  will  each  hind  wheel  make  ? 

13.  Five  times  Harry's  age  plus  4  times  his  age,  minus  6 
times  his  age,  plus  7  times  his  age,  minus  5  times  his  age, 
equals  60  years :   how  old  is  Harry  ? 

14.  A  number  multiplied  by  6,  divided  by  3,  multi- 
plied by  8,  and  divided  by  4,  equals  96 :  what  is  the  num- 
ber? 

WRITTEN  PROBLEMS. 

15.  Divide  486  by  6  ;  by  8  ;  by  9. 

16.  Divide  8408  by  12;  by  24;  by  36. 

17.  Divide  84600  by  900  ;  ^by  12000. 

18.  Divide  412304  by  3600;'  by  303000. 

19.  The  dividend  is  1059984  and  the  divisor  is  306 :  what 
is  the  quotient? 

20.  The  dividend  is  2185750  and  the  quotient  is  250 :  what 
is  the  divisor? 

21.  The  product  is  1123482  and  the  multiplier  is  246: 
what  is  the  multiplicand? 

22.  How  many  passenger  cars,  costing  $2450  each,  can  be 
bought  for  $100450? 

23.  There  are  5280  feet  in  a  mile,  and  the  height  of 
Mount  Everest,  in  Asia,  is  29100  feet :  what  is  its  height  in 
miles? 


DIVISION.  27 

24.  There  are  3600  seconds  in  an  hour :  how  many  hours 
in  738000  seconds  ? 

25.  Divide  the  product  of  480  and  256  by  their  difference. 

DEFINITIONS  AND  PRINCIPLES. 

41.  Division  is  the  process  of  finding  how  many 
times  one  number  is  contained  in  another ;  or,  it  is  the 
process  of  finding  one  of  the  equal  parts  of  a  number. 

The  JJivulend  is  the  number  divided. 

The  Divisor  is  the  number  by  which  the  dividend  is 
divided. 

The  Quotient  is  the  number  of  times  the  divisor  is  con- 
tained, in  the  dividend. 

The  Jletnainder  is  the  part  of  the  dividend  which  is 
left  undivided. 

42.  The  Sign  of  Division  is  a  short  horizontal  line 
between  two  dots,  thus :  -^.  It  is  read  divided  by.  Thus, 
16  -^  4  is  read  16  divided  by  4. 

Division  is  also  expressed  by  writing  the  dividend  above  and  the 
divisor  below  a  short  horizontal  Hne.     Thus,  \^  is  read  18  divided  by  3. 

43.  There  are  two  methods  of  division,  called  Short  Di- 
vision and  Long  Division. 

In  Short  Division^  the  partial  products  and  partial 
dividends  are  not  written,  but  are  formed  mentally. 

In  Long  Division^  the  partial  products  and  partial 
dividends  are  WTitten. 

44. — 1.  One  number  is  contained  in  another  as  many 
times  as  it  must  be  taken  to  produce  it.  Hence,  Division  is 
the  reverse  of  midtiplication. 

2.  One  number  is  contained  in  another  as  many  times  as 
it  can  be  taken  from  it.  Hence,  Division  is  a  brief  method 
of  finding  how  many  times  one  number  can  be  subtracted  from 
another. 


28  COMPLETE  ARITHMETIC. 

45.  Principles. — 1.  The  divisor  and  quotient  are  factors  of 
the  dividend. 

2.  When  division  finds  how  many  times  one  number 
is  contained  in  another,  the  divisor  and  dividend  are  like 
NUMBERS,  and  the  quotient  is  an  abstract  number. 

3.  When  division  finds  one  of  the  equal  parts  of  a  num- 
ber, the  divisor  is  an  abstract  number,  and  the  dividend  and  quo- 
tient are  like  numbers. 

4.  The  m^dtiplying  of  both  divisor  and  dividend  by  the  same 
number  does  not  change  the  value  of  the  quotient. 

5.  The  dividing  of  both  dividend  and  divisor  by  the  same  num- 
ber does  not  change  the  value  of  the  quotient. 

ABBREVIATED  PROCESSES. 
Case  L 

The    IDivisor    lO,    lOO,    lOOO,    etc. 

1.  There  are  10  cents  in  a  dime :  how  many  dimes  in  80 
cents?  120  cents?  240  cents? 

2.  There  are  10  dimes  in  a  dollar :  how  many  dollars  in 
70  dimes?  250  dimes?  2500  dimes? 

3.  There  are  100  cents  in  a  dollar:  how  many  dollars  in 
800  cents?  2400  cents?  7500  cents? 

4.  At  $10  a  barrel,  how  many  barrels  of  flour  can  be 
bought  for  $90?  For  $150? 

5.  At  $100  apiece,  how  many  horses  can  be  bought  for 
$1200  ?     For  $2500  ?     For  $45000  ? 

WRITTEN  PROBLEMS. 

6.  Divide  450  by  10.  7.  Divide  3852  by  100. 

Process.  Process. 

45K)  38152 

45,  Quotient.  38,  Quotient. 

52,  Remainder. 

The  explanation  of  these  processes  is  obvious.  The  cutting  off  of 
the  right-hand  figure  removes  all  the  other  figures  one  place  to  the 
right,  and  thus  decreases  their  value  ten  times.  The  cutting  off  of 
two  figures  removes  the  other  figures  two  places  to  the  right,  and  de- 


DIVISION.  29 

creases  their  value  one  hundred  times.     Tlie  figures  cut  off  denote  the 
remainder. 

8.  Divide  356000  by  100 ;  by  1000. 

9.  Divide  46035  by  100 ;  by"^1000. 

10.  Divide  384602 ;  by  1000 ;  by  10000. 

11.  Divide  95000000  by  10000;  by  1000000. 

46.  Principle.  —  The  removal  of  a  figure  one  order  to  the 
right  decreases  its  value  tenfold. 

47.  Rule.— To  divide  by  10,  100,  1000,  etc.,  Cut  off  as 
many  figures  from  the  right  of  the  dividend  as  there  are  ciphers 
in  the  divisor.      The  figures  cut  off  denote  the  remainder. 

Case  II. 

T'lxe  IDivisor  ending  \vitli  one   or  nriore   Cipliefs. 

12.  There  are  20  quires  of  paper  in  a  ream :  how  many 
reams  in  80  quires?   160  quires? 

13.  There  are  fifty  cents  in  a  half-dollar :  how  many  half- 
dollars  in  150  cents  ?  350  cents  ? 

14.  There  are  60  minutes  in  an  hour :  how  many  hours 
in  240  minutes?  720  minutes? 

15.  A  barrel  of  beef  contains  200  pounds :  how  many 
barrels  will  1200  pounds  make  ?  3600  pounds  ? 

\^7■RITTEN  PROBLEMS. 

16.  Divide  71400  by  3400. 

Process.  First  divide  both  divisor  and  dividend  by 

34100)714100(21  100,  which  is  done  by  cutting  off  the  two 

"^  right-hand  figures.      Then  divide  714,  the 

^  ,          new  dividend,  by  34,  the  new  divisor. 

17.  Divide  58864  by  4500. 

Process.  First  divide  both  dividend  and  divisor  by 
45100  )  588i64  ( 13       100,  which,  in  the  case  of  the  dividend,  leaves 

45  a  remainder  of  64.     Next  divide  588  by  45, 

1^^  leaving  a  remainder  of  3,  which  is  3  hundreds 

-^  since  the  dividend  (588)  is  hundreds.    The  first 

p        •  ^       QRA  remainder  is  64  units  which,  annexed  to  the 

'  3  hundreds,  gives  364,  the  true  remainder. 


30  COMPLETE  ARITHMETIC. 

18.  Divide  63200  by  7900 ;  by  7000. 

19.  Divide  116000  by  2500;  by  4800. 

20.  Divide  172800  by  14400;  by  18000. 

21.  Divide  129600  by  4800;  by  64000. 

48.  Principle. — The  dividing  of  both  divisor  and  dividend 
by  the  same  number  does  not  change  the  value  of  the  quotient. 

49.  Rule. — To  divide  by  a  number  ending  in  one  or  more 
ciphers,  1.  Cut  off  the  ciphers  from  the  right  of  the  divisor, 
and  an  equal  number  of  figures  from  the  right  of  tlie  dividend. 

2.  Divide  the  new  dividend  thus  formed  by  the  new  divisor, 
and  tJie  residt  will  be  tJie  quotient. 

3.  Annex  the  figures  cut  off  from  tlie  dividend  to  tJie  remainder, 
if  tJiere  be  one,  and  the  residt  will  be  the  true  remainder. 

Case  III. 

The  [Divisor  a  convenient  part  of  lO,  lOO,  etc. 

22.  At  3|-  cents  apiece,  how  many  lemons  can  be  bought 
for  90  cents  ?     For  240  cents  ? 

Suggestion. — Since  10  is  3  times  Z\,  multiply  the  dividend  by  3 
and  divide  the  product  by  10. 

23.  At  12^  cents  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  75  cents?     For  225  cents? 

24.  At  16f  cents  a  bushel,  how  many  bushels  of  coal  can 
be  bought  for  150  cents  ?     For  550  cents  ? 

25.  At  $33|  a  head,  how  many  cows  can  be  bought  for 
$200  ?     For  $1200  ? 

^WKITTEN  EXERCISES. 

26.  Divide  4375  by  125.         27.  Divide  13600  by  333^. 

Process.  Process. 

4375  13600 


351000  401800 

35,  Quotient.  40,   Quotient. 

800,  Remainder. 

28.  Divide  6250  by  33^  ;  by  50. 


DIVISION.  31 

29.  Divide  4365  by  250;  by  166|. 

30.  Divide  15300  by  16f ;  by  333 J. 

50.  Principle. — The  multi'plying  of  both  divisor  and  divi- 
dend by  the  same  number  does  not  change  the  value  of  the  quotient. 

51.  Rule. — To  divide  by  a  convenient  part  of  10,  100, 
1000,  etc.,  3fidtiply  the  dividend  by  tJie  number  denoting  how 
many  times  the  divisor  w  contained  in  10,  or  100,  or  1000,  etc., 
and  divide  the  product  by  10,  or  100,  or  1000,  etc. 

Case  IV. 

The  Divisor  a  CoiTiposite  IN'-uixi.'bei'. 

31.  Divide  18315  by  45. 

Process.  Illustrative  Process. 

45  =  5X9  5  )  18315  -r-  45  =^  3663  -~-  9 

5  )  18315  9  )  3663^    9  :=^    407 -f- 1 

9  )  3603  407  -f-    1  =    407 
407,  Quotient. 

Since  45  --  5X9,  the  quotient  obtained  by  dividing  18315  by  5,  is 
9  times  too  large,  and  hence  this  quotient  (3663)  divided  by  9,  is  the 
true  quotient. 

The  process  of  dividing  by  the  factors  of  the  divisor  successively 
is  the  same  in  principle  as  the  division  of  both  dividend  and  divisor 
by  these  factors  successively,  which  (Art.  48)  does  not  change  the 
value  of  the  quotient.     See  "  Illustrative  Process." 

32.  Divide  58636  by  28  ;  by  77. 

33.  Divide  13328  by  49  ;  by  56  ;  by  70. 

34.  Divide  31360  by  64;  by  70;  by  81. 

35.  Divide  3687  by  64. 

Process. 
2  )_3687        64  =  2  X  8  X  4 

8  )1843 ....  1  (1st  Kern.)  = 1 

4  )^  ....  3  (2d     "      )  =  3  X  2  =  . . . .  6 

57 2  (3d     "      )  =  2  X  8  X  2  =  32 

True  Kemainder  =  39 

A  unit  of  the  first  quotient  equals  2  units  of  the  dividend,  and 
hence  the  second  remainder  (3)  equals  3X2  units  of  the  dividend. 


32  COMPLETE  ARITHMETIC. 

A  unit  of  the  second  quotient  equals  8  units  of  the  first  quotient,  and 
hence  the  third  remainder  (2)  equals  2X8  units  of  the  second  quo- 
tient =  2X8X2  units  of  the  dividend.  Hence  the  first  remainder 
is  1 ;  the  second  6 ;  the  third  32 ;  and  the  total,  or  true  remainder,  39. 

Note. — The  teacher  can  illustrate  this  process  by  considering  the 
dividend  (3687)  'pints.  The  first  quotient  will  be  quarts,  the  second 
pecks,  and  third  bushels,  and  the  first  remainders  will  be  1  pt.,  the 
second,  3  qt.,  and  the  third,  2  pk.     1  pt.  +  3  qt.  -\-  2  pk.  =  39  pt. 

36.  Divide  34567  by  63 ;  by  72. 

37.  Divide  120473  by  56;  by  81. 

38.  Divide  400671  by  64 ;  by  77. 

39.  Divide  346000  by  bb  ;  by  96. 

40.  Divide  47633  by  90;  by  110. 

52.  Pkinciple. — The  division  of  both  divisor  and  dividend 
by  tJw  same  number  does  not  change  the  value  of  the  quotient. 

53.  Rule. — To  divide  by  a  composite  number,  1.  Resolve 
the  divisor  into  convenient  factors ;  divide  the  dividend  by  one  of 
these  factors,  the  quotient  thus  obtained  by  another,  and  so  on 
until  all  the  factors  are  used  as  divisors.  The  last  quotient  will 
be  the  true  quotient. 

2.  Multiply  each  remahider,  except  the  first,  by  all  the  divisors 
preceding  its  own.  TJie  sum  of  these  products  and  the  first  re- 
mainder will  be  the  true  remainder. 


SECTION   VII. 

PROPERTIES  OF  NUMBERS. 
PRIME  AND  COMPOSITE  NUMBERS  AND  FACTORS. 

Note. — The  terms  number,  divisor,  and  factor,  used  in  this  section, 
denote  integral  numbers. 

1.  What  two  numbers  besides  itself  and  1  will  exactly 
divide  10?  21?  35?  63?  77? 

2.  What  numbers  besides  itself  and  1  will  exactly  divide 
7?  11?  17?  23?  37?  41? 


PROPERTIES  OF  NUMBERS.  33 

3.  What  numbers  will  exactly  divide  15?  13?  28?  29? 
42?  43? 

Note. — Since  every  integer  is  exactly   divisible  by   itself  and  1, 
these  divisors  need  not  be  given. 

4.  What  numbers  will 'exactly  divide  30?  31?  45?  53? 
56?  67?  65? 

5.  Name  all  the  prime  numbers  between  0  and  20 ;    30 
and  50. 

6.  Name  all  the  composite  numbers  between  20  and  30 ; 
50  and  70. 

7.  What  are  the  prime  divisors  of  6  ?  15?  18?  21?  30? 
45?  50?  54? 

8.  What  are  the  prime  factors  of  12?   24?   35?  39?  42? 

9.  What  are  the  prime  factors  of  27?  36?  49?  56?  63? 
66?  72?  84? 

10.  Of  what  numbers  are  2  and  5  prime  factors?   2,  3, 
and  5  ?    2,  5,  and  7  ?    3,  5,  and  7  ? 

11.  Of  what  numbers  are  2,  2,  and  3  prime  factors?  2,  3, 
3,  and  5  ?   2,  3,  5,  and  7  ? 

12.  What  prime  factor  is  common  to  9  and  12?    15  and 
25?  18  and  30?   21  and  28? 

13.  What  prime  factor  is  common  to  24  and  27  ?   35  and 
42?  44  and  77?    35  and  50?   63  and  70? 

WRITTEN  EXERCISES. 

14.  What  are  the  prime  factors  of  126? 

Process.  Divide  126  by  2,  a  prime  divisor ;    next 

2 )  126  divide    the    quotient    63    by    3,  a    prime 

3  -j  g3  divisor,  and  then  divide  the  quotient  21 

3  )'2l  by  3,  a  prime  divisor.     The  prime  factors 

7  are  2,  3,  3,  and  7. 

126  =  2  X  3  X  3  X  7. 

What  are  the  prime  factors  of 

15.  160?         18.  325?         21.  462?  24.    748? 

16.  175?        19.  330?        22.  490?  25.    693? 
17.256?        20.420?        23.  .594?  26.1155? 


34  COMPLETE  ARITHMETIC. 

What  prime  factors  are  common  to 

27.  45  and    63?  30.  200  and  250? 

28.  50  and    80?  31.  175  and  325? 

29.  96  and  256?  32.   144  and  180? 

DEFINITIONS,  PRINCIPLES,  AND  RULES. 

54.  The  Divisor  of  a  number  is  any  number  that  will 
exactly  divide  it. 

55.  Numbers  are  either  Prime  or  Composite. 

A  Prime  Number  has  no  divisor  except  itself  and 
one. 

A  Composite  Wuinber  has  other  divisors  besides 
itself  and  one. 

Every  composite  number  is  the  product  of  two  or  more  numbers, 
called  factors. 

56.  Two  or  more  numbers  are  prime  to  each  other,  or  rela- 
tively prijne,  when  they  have  no  common  divisor  except  1. 
Thus,  9  and  16  are  prime  to  each  other. 

All  prime  numbers  are  prime  to  each  other.  Composite  numbers 
may  be  relatively  prime,  as  9  and  10 ;  16  and  25. 

57.  A  Factor  of  a  number  is  its  divisor. 

A  Prime  Factor  of  a  number  is  its  prime  divisor. 

The  terms  divisor  and  factor  differ  only  in  their  use,  the  former 
implying  division  and  the  latter  multiplication.  A  divisor  or  factor  of 
a  number  is  also  called  its  measure. 

58.  When  a  number  is  a  factor  of  each  of  two  or  more 
numbers,  it  is  called  their  Comm^on  Factor,  Thus,  5 
is  a  common  factor  of  15  and  20. 

59.  Whole  numbers  are  either  Even  or  Odd. 

An  Even  N'Um^ber  is  exactly  divisible  by  2 ;  as,  2,  4, 
6,  8,  10,  12,  etc. 

An  Odd  Number  is  not  exactly  divisible  by  2 ;  as,  1 , 
3,  5,  7,  9,  11,  13,  etc. 


CANCELLATION.  35 

All  the  even  numbers  except  2  are  composite.  Some  of  the  odd 
numbers  are  composite  and  others  are  prime. 

60.  Principles. — 1.  A  factor  of  a  number  is  a  factor  of 
any  number  of  times  that  number. 

2.  A  common  factor  of  two  or  more  numbers  is  a  factor  of 
their  sum. 

3.  A  composite  number  is  the  product  of  all  its  prime  factors. 

4.  If  a  composite  number  composed  of  two  factors  be  divided 
by  one  fador,  the  quotient  will  be  the  other  factor. 

5.  If  any  composite  number  be  divided  by  a  factor,  or  by  the 
product  of  any  number  of  its  factors,  the  quotient  will  be  the 
product  of  the  remaining  factors. 

61.  KuLES. — 1.  To  resolve  a  composite  number  into  its 
prime  factors,  Divide  it  by  any  prime  divisor,  and  the  quo- 
tient by  any  prime  divisor,  and  so  continue  until  a  quotient  is 
obtained  which  is  a  prime  number.  The  several  divisors  and 
tJie  last  quotient  are  the  prime  factors. 

2.  To  find  the  common  factors  of  two  or  more  numbers, 
Resolve  Uie  given  numbers  into  their  prime  factors  and  select  the 
factors  which  are  found  in  all  the  numbers. 

CANCELLATION. 

33.  Divide  the  product  of  4,  7,  9,  and  12  by  the  product 
of  4,  7,  and  9. 

Process.  Instead  of  forming  the  prod- 
Dividend,  ^  X  y^  X  0  X  12  "^^^'  indicate  the   multiplica- 
.            — VZTirZT^        ^12.  tion  by  the  proper  sign,  and 
Divisor,           /i  X  ^  X  0  V     *u     J-   •              J          ^1 
J           f    /\  i>  /\  t  write  the  divisor   underneath 

the  dividend.  Since  dividing  both  dividend  and  divisor  by  the  same 
number  does  not  affect  the  value  of  the  quotient  (Art.  48),  divide 
each  by  4,  7,  and  9.  This  may  be  done  by  canceling,  as  indicated  in 
the  process.     The  quotient  is  12. 

34.  Multiply  4  X  7  by  12,  and  divide  the  product  by  4 
times  12. 

35.  Divide  6  X  8  X  20  by  4  X  20. 

36.  Divide  5  X  7  X  11  X  13i  by  7  X  13f 


36  COMPLETE  ARITHMETIC. 

37.  Divide  12  X  16  X  2$  by  9  X  24  X  21. 

Process.  Since  dividing  the  factor 

4  of  a  number  divides  the  num- 

J:^  X  10  X  ;^^  _  8  X  4      32  .  her,  cancel   12  in  the  divi- 

9  X  M  X  M  ~  9  X  S~Y7~^^''^        dend  and   divide  24  in   the 
;2  3  divisor  by  12,  giving  2.    Can- 

cel the  2  and  divide  16  in  the 
dividend  by  2,  giving  8.  Divide  the  28  in  the  dividend  and  21  in 
the  divisor,  each  by  7,  giving  4  and  3.  The  uncanceled  factors  of 
the  divisor  are  8  and  4,  and  those  of  the  dividend  are  9  and  3.  The 
quotient  is  32  -r-  27  =:^  lo^. 

38.  Divide  24  X  27  X  12^  by  18  X  54  X  50. 

39.  Divide  28  X  30  X  100  by  21  X  15  X  33^. 

40.  40  X  22  X  35  X  16|  -^  20  X  44  X  50  X  49  :^  what? 

41.  A  farmer  exchanged  12  barrels  of  apples,  each  con- 
taining 3  bushels,  at  75  cts.  a  bushel,  for  25  sacks  of  pota- 
toes, each  containing  2  bushels :  how  much  did  the  potatoes 
cost  a  bushel? 

42.  If  9  men  can  do  a  piece  of  work  in  15  days,  working 
10  hours  a  day,  how  many  men  can  do  it  in  20  days,  work- 
ing 8  hours  a  day  ? 

DEFINITIONS,  PRINCIPLES,  AND  RULE. 

62.  Cancellatiou  is  the  omission  of  one  or  more  of 
the  factors  of  a  number.  It  is  used  when  both  dividend  and 
divisor  contain  one  or  more  equal  factors,  to  abbreviate  the 
process  of  division. 

63.  Principles. — 1.  The  canceling  of  one  of  the  factors  of 
a  number  divides  the  number  by  the  factor  canceled. 

2.  Canceling  equal  factors  of  both  dividend  and  divisor 
divides  them  by  the  same  number,  and  hence  does  not  change  the 
value  of  the  quotient. 

3.  Dividing  one  of  the  composite  factors  of  a  product  divides 
the  product. 

64.  Rule. — Indicate  the  multiplications  by  the  proper  sign, 
and  write  the  divisor  underneath  the  dividend.      Cancel  the  fac- 


COMMON    DIVISOR.  37 

tors  common  to  both  dividend  and  divisor,  and  divide  the  prod- 
uct of  the  factors  left  in  the  dividend  by  the  product  of  those 
left  in  the  divisor. 

]Nf  OTE. — When  all  the  expressed  factors  of  either  dividend  or  divisor 
are  canceled,  1  remains  as  a  factor. 

GREATEST  COMMON  DIVISOR. 

1.  What  are  the  divisors  of  15?  28?  45?  53?  75?  90? 
91 ?  108  ? 

2.  What  is  a  common  divisor  of  15  and  35?  42  and  56? 
63  and  72  ?  64  and  80  ? 

3.  What  is  a  common  divisor  of  27  and  36?  18,  30,  and 
42?  36,  54,  and  72? 

4.  What  is  the  greatest  number  that  will  exactly  divide 
32  and  48?  45  and  90?  60  and  96? 

5.  What  is  the  greatest  common  divisor  of  36  and  60  ? 
45,  60,  and  75?  18,  54,  and  90? 

6.  What  is  the  greatest  common  divisor  of  24,  48,  and 
72?  16,  48,  and  80?  20,  31,  and  45? 

7.  Show  that  every  common   divisor  of  12   and  16  is  a 
divisor  of  28,  their  sum. 

8.  Show  that  a  common  divisor  of  any  two  numbers  is  a 
divisor  of  their  sum. 

9.  Show  that  every  common   divisor  of  16  and  28  is  a 
divisor  of  12,  their  difference. 

10.  Show  that  a  common  divisor  of  any  two  numbers  is  a 
divisor  of  their  difference. 

WKITTEN   EXERCISES. 

11.  What  is  the  greatest  common  divisor  of  126  and  210? 

Process  by  Factoring.         Resolve  126  and  210  into  their  prime 
126  =  ^X$X3X/J'  factors.     Since  every  divisor  of  a  num- 

2]^0  =  0VJX5X'^  ^^^  ^^  ^  prime  factor,  or  the  product  of 

2X3X7=42,  CCA  *™  7  7;'f  p'™^ ^Tr' ""*  ''"^' 

'  uct  oi   all  the  prime   lactors  common 

to  126  and  210  will  be  their  greatest  common  divisor. 


38  COMPLETE  ARITHMETIC. 

What  is  the  greatest  common  divisor  of 

12.  60  and    84?  15.   112,  140,  and    168? 

13.  63  and  126?  16.     84,  126,  and    210? 

14.  144  and  192?  17.  128,  256,  and  1280? 

18.  What  is  the  greatest  common  divisor  of  288  and  528  ? 

Process  by  Dividing.  Divide  528  by  288,  and  288  by  the 

288  )  528  ( 1  first  remainder  240,  and  240  by  the  see- 

288  ond   remainder   48 ;    and,    there   being 

240  )  288  (  1  no  remainder,  48  is  the  greatest  com- 

^  mon  divisor  of  288  and  528. 

48  )  240  (  5  Since  48,  which  is  the  greatest  divisor 

of  itself,  is  a  divisor  of  240,  it  is  the  cpxat- 
48  =  G.aD.of288and528.  ,,,  common  divisor  of  48  ™d  240.  Since 
48  is  a  divisor  of  both  48  and  240,  it  is  a  divisor  of  288,  their  swm, 
and  since  the  greatest  common  divisor  of  two  numbers  is  a  divisor  of 
their  difference,  48  is  the  greatest  common  divisor  of  240  and  288. 
Since  48  is  the  greatest  common  divisor  of  240  and  288,  it  is  a  divisor 
of  528,  their  sum  ;  and  since  the  greatest  common  divisor  of  two  num- 
bers is  a  divisor  of  tlieir  difference,  48  is  the  greatest  common  divisor 
of  288  and  528. 

Note. — Let  the  pupil  show,  in  like  manner,  that  the  last  divisor, 
in  the  solution  of  each  of  the  following  problems,  is  the  greatest  com- 
mon divisor  required. 

What  is  the  greatest  common  divisor  of 

27.  S260  and    $416? 

28.  $1815  and  $3465? 

29.  21451b.  and  34711b.? 

30.  175,  225,  and    275? 

31.  240,  360,  and    480? 

32.  144,  216,  and    648? 

33.  140,  308,  and    819  ? 

34.  240,  336,  and  1768? 

35.  What  is  the  greatest  common  divisor  of  1065,  1730, 
and  2845? 

36.  What  is  the  greatest  common  divisor  of  156,  585, 
442,  and  1287? 

37.  What  is  the  greatest  common  divisor  of  2731  and  3120? 


19. 

196  and  1728? 

20. 

336  and  576? 

21. 

407  and  888? 

22. 

326  and  807  ? 

23. 

756  and  1764? 

24. 

1064  and  1274? 

25. 

768  and  5184? 

26. 

741  and  1938? 

LEAST  COMMON  MULTIPLE.  39 


DEFINITIONS,  PRINCIPLES,  AND  RULES. 

65.  A  Divisor  of  a  number  is  a  number  that  will  ex- 
actly divide  it. 

A  Conifnon  Divisor  of  two  or  more  numbers  is  a 
number  that  will  exactly  divide  each  of  them. 

The  Greatest  Conimou  Divisor  of  two  or  more 
numbers  is  the  greatest  number  that  will  exactly  divide  each 
of  them. 

66.  Principles. — 1.  Every  prime  factor,  and  every  product 
of  any  two  or  more  prims  factors  of  a  number,  is  a  divisor  of 
tliat  number.     Conversely, 

2.  Every  divisor  of  a  number  is  a  prime  factor,  or  the  product 
of  two  or  more  of  its  prime  factors. 

3.  The  product  of  all  tlie  prime  factors  common  to  two  or  more 
numbers  is  their  greatest  common  divisor. 

4.  The  divisor  of  a  number  is  a  divisor  of  any- number  of 
times  that  number. 

5.  A  common  divisor  of  two  numbers  is  a  divisor  of  their 
sum,  or  of  their  difference. 

6.  Any  common  divisor  of  eitJier  of  two  numbers  and  their 
difference  is  a  common  divisoi'  of  the  tivo  numbers. 

67.  Rules. — ^1.  To  find  the  greatest  common  divisor  of  two 
or  more  numbers  by  factoring,  Resolve  the  given  ^lumbers  into 
tlieir  prime  factors,  and  select  the  factors  which  are  common. 
The  product  of  the  common  factors  will  be  the  greatest  common 
divisor. 

2.  To  find  the  greatest  common  divisor  of  two  numbers 
by  division,  Divide  the  greater  number  by  the  less,  and  the 
divisor  by  the  remainder,  and  the  second  divisor  by  the  second  re- 
mainder, and  so  on,  until  there  is  no  remainder.  The  last  divisor 
will  be  the  greatest  common  divisor. 

Note. — When  there  are  three  or  more  numbers,  first  find  the  great- 
est common  divisor  of  two  of  them,  and  then  the  greatest  common 
divisor  of  this  G.  C.  D.  and  a  third  number,  and  so  on. 


40  COMPLETE   ARITHMETIC. 

LEAST   COMMON  MULTIPLE. 

1.  What  number  will  16  exactly  divide?    25?    30?    45? 
Note. — A  number  will  exactly  divide  its  multiple. 

2.  What  number  is  a  multiple  of  15?  24?  32?  54?  75? 
100?    120?   150?   200? 

3.  How  many  multiples  has  every  number? 

4.  What  number  will  8  and  10  both  exactly  divide?  9 
and  12?  20  and  25? 

5.  What  number  is  a  common  multij^le  of  5  and  12  ?  15 
and  30?  25  and  50? 

6.  How  many  common  multiples  have  two  or  more  num- 
bers? 

7.  What  is  the  least  number  that  7  and  8  will  both  ex- 
actly divide?  9  and  12?  20  and  30?  25  and  75? 

8.  What  number  is  the  least  common  multiple  of  7  and 
10?  12  and  18?  8,  12,  and  16? 

9.  How  many  least  common  multiples  have  two  or  more 
numbers  ? 

10.  Show  that  all  the  prime  factors  of  a  number  are 
factors  of  its  multiple,  and,  conversely,  that  a  number  con- 
taining all  the  prime  factors  of  another  number  is  its  mul- 
tiple. 

WKITTEN    EXERCISES. 

11.  What  is  the  least  common  multiple  of  12,  18,  and  30? 

Process  by  Factoring.  Resolve  tlie  numbers  into 

12  =  ^  X  ^  X  3  their  prime  factors,  and  select 

.r, 9  V  ^  V  S  ^^^   *^^   different    factors,   re- 

peating each  as  many  times  as 

/\      /\  5(/ j^.    jg    found   in  any  number. 

2X2X3X3X5  =  180,  L.  C.  M.        The  factor  2  is  found  twice  in 

12;  the  factor  3,  twice  in  18; 
and  the  factor  5,  once  in  30.  The  product  of  2X2X3X3X5  is 
the  least  common  multiple  required,  since  it  is  the  least  number  which 
contains  all  the  prime  factors  of  12,  18,  and  30. 


Process  by 
2)12        15 

Division. 
42        70 

3)6 

15 

21 

35 

5)2 

5 

7 

35 

7)2 

1 

7 

7 

2           1 
2X3X5X7X 

1 
2  = 

1 
420,  L.  a  M. 

LEAST   COMMON   MULTIPLE.  41 

What  is  the  least  common  multiple  of 

12.  8,  12,     20?  16.     18,     24,     72,     48? 

13.  9,  21,     42?  17.     15,    35,     70,  105? 

14.  32,  48,     80?  18.     25,     75,  100,  150? 

15.  27,  54,  108?  19.  $16,  $40,  $60,  $72? 

20.  What  is  the  least  common  multiple  of  12,  15,  42,  70? 

Find  all  the  prime  factors 
by  dividing  the  given  num- 
bers by  any  prime  number 
that  will  exactly  divide  two  or 
more  of  them,  thus  :  Dividing 
by  2,  it  is  found  to  be  a  prime 
factor  of  12,  42,  and  70.  Write 
the  quotients  with  the  15  un- 
derneath. Dividing  by  3,  it 
is  found  to  be  a  prime  factor  of  6,  15,  and  21,  and  hence  it  is  a  prime 
factor  of  12,  15,  and  42.  Dividing  by  5,  it  is  found  to  be  a  prime 
factor  of  5  and  35,  and  hence  of  15  and  70.  Dividing  by  7,  it  is 
found  to  be  a  prime  factor  of  7  and  7,  and  hence  of  42  and  70.  The 
remaining  quotient  2  is  a  prime  factor  of  12. 

Hence,  all  the  prime  factors  of  12,  15,  42,  and  70  are  2,  3,  5,  7,  and 
2,  and  since  the  product  of  these  several  prime  factors  (2  X  3  X  5  X 
7X2  =  420)  is  the  least  number  that  contains  each  of  them,  it  is  the 
least  common  multiple  of  12,  15,  42,  and  70. 

What  is  the  least  common  multiple  of 

21.  12,  18,  30?  26.  30,  45,  48,  80,  120? 

-22.     8,  28,  70?  27.  16,  30,  40,  50,     75? 

23.  9,  20,  15,  36?         28.  15,  27,  35,  42,     70? 

24.  15,  24,  25,  30?         29.  8,  28,  20,  24,  32,  48? 

25.  18,  21,  27,  36?         30.  2,  3,  4,  5,  6,  7,  8,  9? 

DEFINITIONS,  PRINCIPLES,  AND  RULES. 

68.  A  3Iultiple  of  a  number  is  any  number  which  it 
will  exactly  divide. 

Note. — Every  number  is  an  exact  divisor  of  its  product  by  an 
integer. 

C.Ar.— 4 


42  COMPLETE  ARITHMETIC. 

A  Common  Multiple  of  two  or  more  numbers  is 
any  number  which  each  of  them  will  exactly  divide. 

The  Least  Common  Multiple  of  two  or  more 
numbers  is  the  least  number  which  each  of  them  will  ex- 
actly divide. 

Note. — The  following  definitions  may  be  preferred  by  some 
teachers : 

A  Muhiple  of  a  number  is  the  product  arising  from  taking  it  one 
or  more  times. 

A  Common  Multiple  of  two  or  more  numbers  is  a  number  which  is 
a  multiple  of  each  of  them. 

The  Least  Common  Multiple  of  two  or  more  numbers  is  the  least 
number  which  is  a  multiple  of  each  of  them, 

69.  Principles. — 1.  Every  multiple  of  a  number  contains 
all  its  'prime  factors. 

2.  A  common  multiple  of  two  or  more  numbers  contains  all 
their  prime  factors. 

3.  The  least  common  multiple  of  two  or  more  numbers  con- 
tains all  their  prime  factors,  and  no  other  factors. 

4.  The  least  common  multiple  of  two  or  more  numbers  con- 
tains each  of  their  prime  factors  the  greatest  number  of  times  it 
occurs  in  either  number. 

70.  Rules. — 1.  To  find  the  least  common  multiple  of 
two  or  more  numbers  by  factoring,  Resolve  each  of  the  num- 
bers into  its  prime  factors,  and  then  select  all  the  different  factors, 
taking  each  the  greatest  number  of  times  it  is  found  in  any 
number.  The  product  of  the  different  factors,  thus  selected,  will 
he  the  least  common  multiple. 

2.  To  find  the  least  common  multiple  of  two  or  more 
numbers  by  division,  Write  the  numbers  in  a  line,  and  divide 
by  any  prime  divisor  of  two  or  more  of  them,  tvriting  the  quotients 
and  the  undivided  numbers  underneath.  Divide  these  resulting 
numbers  by  any  prime  divisor  of  two  or  more  of  them,  and  so 
proceed  until  no  two  of  the  resulting  numbers  have  a  common 
prime  divisor.  The  product  of  the  divisors  and  tJie  last  result- 
ing numbers  will  he  the  least  common  multiple  required. 

js^OTE.— If  no  two  of  the  given  numbers  have  a  common  divisor, 
their  product  will  be  the  least  common  multiple 


FRACTIONS. 


43 


SECTION    VIII. 

FRACTIONS 


FOURTHS 


HALVES 


SIXTHS 


NUMERATION  AND  NOTATION. 

1.  If  a  single  thing  be  divided  into  two  equal  pieces,  what 
part  of  the  whole  will  one  piece  be  ? 

2.  If  a  single  thing  be  divided  into  four  equal  pieces, 
what  part  of  the  whole  will  one  piece  be?  Two  pieces? 
Three  pieces? 

3.  How  many  halves  in  a  single  thing  or  unit?  How 
many  fourths  ? 

4.  Which  is  the  greater,  one  half  or  one  fourth  of  a 
unit?     How  many  fourths  in  one  half? 

5.  What  is  meant  by  one  third  of  a  unit?  Two  thirds? 
One  sixth?    Three  sixths?     Two  fifths?     Four  fifths? 

71.  Such  parts  of  a  unit  as  one  half,  two  thirds,  three 
fourths,  etc.,  are  called  Fractions.  A  fraction  may  be  ex- 
pressed in  figures  by  writing  the  figure  denoting  the  number 
of  equal  parts,  into  which  the  unit  is  divided,  below  a  short 


44  COMPLETE  ARITHMETIC. 

horizontal  line  (^),  and  the  figure  denoting  the  number  of 
equal  parts  taken,  above  the  same  line  (^).  Thus,  f  ex- 
presses five  sixths  of  a  unit. 

6.  What  does  f  express?  What  does  the  figure  7,  below 
the  line,  denote?     The  figure  5,  above  the  line? 

Bead  the  following  fractions,  and  tell,  in  each  case,  what 
each  figure  denotes : 


7.  f 

10.  1 

13.  J, 

16.  A 

8.  i 

U.  f 

14.  A 

17.  U 

9.  f 

12.  i 

15.  A 

18.  U 

Write  the  following  fractions  in  figures : 
(19)                           (20)  (21) 

Two  fifths ;  Seven  twelfths ;  Twenty-four  fortieths ; 

Seven  ninths;         Ten  thirteenths ;  Thirty-five  fiftieths; 

Ten  ninths.  Twenty  seventeenths.  Forty  fifty-fifths. 

22.  Is  the  fraction  f  greater  or  less  than  1?     Why? 

23.  Is  I  greater  or  less  than  a  unit?     Why? 
Compare  the  value  of  each  of  the  following  fractions  with 

a  unit  or  1 : 

24.  f  26.  Y  28.     I  30.  {-} 

25.  I  27.  j\  29.  H  31.  U 

32.  Deduce  from  the  above  examples  a  general  statement 
of  the  value  of  fractions  as  compared  with  a  unit  or  1. 

DEFINITIONS  AND  PRINCIPLES. 

72.  A  Fraction  is  one  or  more  of  the  equal  parts  of 
a  unit. 

The  unit  divided  is  called  the  Unit  of  the  Fraction  ;  and  one  of  the 
equal  parts,  into  which  it  is  divided,  is  called  a  Fractional  Unit.  An 
integer  is  composed  of  integral  units,  and  a  fraction  of  fractional 
units. 

73.  A  Common  Fraction  is  expressed  in  figures  by 

two  numbers,  one  written  over  the  other^  with  a  line  between 

them. 

Note.  —  Decimal  fractions  are  a  varietv  of  common  fractions. 
(Art.  112.) 


FRACTIONS.  45 

The  number  above  the  line  is  called  the  Numerator ;  and 
the  one  below  the  line,  the  Denominator. 

The  Denominator  of  a  fraction  denotes  the  number 
of  equal  parts  into  which  the  unit  is  divided. 

The  Numerator  of  a  fraction  denotes  the  number  of 
equal  parts  taken. 

The  numerator  and  denominator  are  called  the  Terms 
of  the  fraction. 

74.  Principle. — The  value  of  a  fraction  is  less  than  1 
ivhen  its  numerator  is  less  than  its  denominator ;  equal  to  1  when 
its  numerator  equals  its  denominator ;  and  more  Uian  1  when  its 
numerator  is  greater  than  its  denominator. 

75.  Common  Fractions  are  Proper  or  Improper. 

A  JProper  Fraction  is  one  whose  numerator  is  less 
than  its  denominator ;  as,  |,  |. 

An  Improx^er  Fraction  is  one  whose  numerator  is 
equal  to  or  greater  than  its  denominator. 

The  Talue  of  a  proper  fraction  is  less  than  one ;  and  the  value  of 
an  improper  fraction  is  equal  to  or  greater  than  one,  and  hence  it  is 
regarded  as  not  properly  the  fraction  of  a  unit. 

76.  Fractions  are  Simple^  Compound ,  or  Complex. 

A  Simple  Fraction  is  a  fraction  not  united  with  an- 
other, and  both  of  whose  terms  are  integral ;  as,  |. 

A  Compound  Fraction  is  a  fraction  of  a  fraction ; 

as,  I  off;  |of  3f 

A  Complex  Fraction  is  one  having  a  fraction  in 

A  Mixed  Wumber  is  an  integer  and  a  fraction  united ; 
as,  5 J,  16|. 

77.  The  fraction  |  may  be  considered  as  expressing  3 
fifths  of  1  unit,  or  1  fifth  of  3  units ;  and  hence  the  numer- 


2 

one  or  both  of  its  terms ;  as,  h  3'  f'  qi 


46  COMPLETE  ARITHMETIC. 

ator  of  a  fraction  may  denote  the  number  of  units  to  be 
divided,  and  the  denominator  the  number  of  parts  into 
which  the  numerator  is  to  be  divided.  Thus,  f  may  be  read 
5  sixths,  or  1  sixth  of  5,  or  5  divided  by  6.     Hence, 

A  frojction  may  express  an  unexecuted  division,  the  numer- 
ator denoting  the  dividend,  tJie  denominator  the  divisor,  and  the 
fraction  itself  the  quotient. 

EEDUCTION  OF  FRACTIONS. 
Case  I. 

"Wh-ole    or   IVTixed    ^N'mnbers    reduced  to    Improper 
Fractions. 

1.  How  many  thirds  in  an  apple?    In  4  apples?  7  ap- 
ples? 10  apples?  20  apples? 

2.  How  many  fifths  in  3  melons  ?     In  five  melons  ?    8 
melons?    12  melons?    15  melons? 

3.  How  many  sixths  in  1  ?   5?   8?   12?   20? 

4.  How  many  fourths  of  an  inch  in  2  and  1  fourth  inches? 
In  3|  inches  ?   6f   inches  ?  30^  inches  ? 

5.  How  many  fifths  in  3|?    4|?    12f?   16f? 

6.  How  many  tenths  in  5xV?    8fV?    12  ,\?    15^? 

WRITTEN   PROBLEMS. 

7.  Reduce  225  to  sevenths.         225 1  to  sevenths. 

Process.  Process. 

225  225^ 

7  7 

1575,  Ans,  l^P,  Ans, 

7  7 

8.  Reduce  324  to  ninths.     324|^  to  ninths. 

9.  Reduce  48fi  to  15ths.     65||  to  15ths. 

10.  Reduce  54^V  to  20ths.     135|J  to  30ths. 

11.  Reduce  63  j\  to  an  improper  fraction. 

12.  Reduce  74  {i  to  an  improper  fraction. 

13.  Reduce  206/^  to  an  improper  fraction. 


REDUCTION  OF  FRACTIONS.  47 

14.  Reduce  145/^  to  an  improper  fraction. 
Reduce  to  an  improper  fraction, 

15.  137^  17.  eOO^f  19.  208^ 

16.  408^0  18.  365^  20.  607^% 

78.  Rules. — 1.  To  reduce  an  integer  to  a  fraction,  3ful' 
tiply  the  integer  by  the  given  denominator,  and  write  the  denomi- 
nator under  the  prodmit. 

2.  To  reduce  a  mixed  number  to  a  fraction,  Multiply  the 
integer  by  the  denominator  of  the  fraction,  to  the  product  add 
tJie  numerator y  and  write  the  denominator  under  the  residt. 

Case   II. 

IiTiproper    jFractions   reclncecl    to  "Whole    or    IVEixed. 

21.  How  many  dollars  in  8  half-dollars?  16  half-dollars? 
30  half-dollars  ? 

22.  How  many  pints  in  9  thirds  of  a  pint?  15  thirds  of 
a  pint  ?   33  thirds  of  a  pint  ? 

23.  How  many  days  in  20  fifths  of  a  day?  35  fifths  of  a 
day?   42  fifths  of  a  day? 

24.  How  many  units  in  36  ninths?  63  ninths?  75  ninths? 

25.  How  many  units  in  -2/-?   A^?   -%^-?   V? 

26.  How  many  units  in  fl?    f^?    V^?   %^-'i 

WKITTEN  PKOBLEMS. 

27.  Reduce  ^^~  to  a  whole  number. 
Process  :  ^6^^.=  256  h-  16  =  16,  Am. 

28.  Reduce  -\^/  to  a  mixed  number. 
Reduce  to  a  whole  or  a  mixed  number, 

29.  W         32.     -«#         35.  m^         88.     -V^- 

30.  -W-         33.     -V/         36.  ^-^^A         39.     ^ffs 

31.  %5.         34  1128  37,  426_a         40.  i^j. 

79.  Rule. — To  reduce  an  improper  fraction  to  an  integer 
or  mixed  number,  Divide  the  numerator  of  the  fraction  by  the 
denominator. 


48  COMPLETE   ARITHMETIC. 

Case  III. 

Siinple  Frac^tions  redxiced  to  Lowest  Terms. 

41.  How  many  fourths  of  an  inch   in  2  eighths  of  an 
inch  ?     In  4  eighths  ?   6  eighths  ? 

42.  How  many  sixths  in  2  twelfths  ?     In  4  twelfths  ?    8 
twelfths?    10  twelfths? 

43.  How  many  sevenths   in  4  fourteenths?     In  6  four- 
teenths?   8  fourteenths?    12  fourteenths? 

44.  How  many  eighths  in  3% ?    ig?   i|?   f-g? 

45.  How  many  tenths  in  3«o?   M  ?   IV   W 

46.  Reduce  f-^,  ^f ,  ^  each  to  fifths. 

47.  Reduce  J^,  f f,  Jf,  and  -g-J  each  to  sixths. 

48.  Divide  both  terms  of  -f^  by  3,  and  show  that  the  value 
of  the  fraction  is  not  changed. 

49.  Show  that  the  division  of  both  terms  of  any  fraction 
by  the  same  number  does  not  change  its  value. 

WRITTEN   PROBLEMS. 

50.  Reduce  -f  Jf  to  its  lowest  terms. 

Pkocess.  Divide  both  terms  of  |£^-  by  5,  re- 

105  -^  0  ^  21     21jj-7^3        ducing    it   to   fi;    then   divide    both 

140-^5    "28     28 -r- 7       4        terms  of   f|  by  7,  reducing  it  to  f. 

Since  f  can  not  be  reduced  to  smaller 

Or :   — —  ^  - '  Ans.  or  lower  terms,  it  is  in  its  lowest  terms. 

140  ^35       4  Qj.^  ^j^.^g  ^^^1^  ^^j,^^  ^f  ^05  by  35, 

their  greatest  common  divisor,  reducing  the  fraction  to  |. 
Reduce  to  lowest  terms, 

51.  t'^  54.  IIJ  57.     m  60.  ^VA 

52.  m  55.  m  58.  ,V^  61.  T%V 

53.  Ill  56.  m  59.  tWf  62.  |^^ 

63.  Express  the  quotient  of  195  divided  by  105  in  its 
simplest  form.  Ans.  -^-^. 

64.  Express  the  quotient  of  462  divided  by  441   in  its 
simplest  form.  128  --  256.  360  ^  288. 


REDUCTION  OF  FRACTIONS.  49 

65.  Express  the  quotient  of  576  ^-  432  in  its  simplest 
form.     216  ^  324.     828  --  506. 

DEFINITIONS,  PRINCIPLE,  AND  RULES. 

80.  A  fraction  is  reduced  to  lower  terms  when  it  is  changed 
to  an  equivalent  fraction  with  smaller  terms. 

A  fraction  is  in  its  lowest  terms  when  its  terms  are  prime 
to  each  other. 

81.  Principle. — The  division  of  both  terms  of  a  fraction  by 
the  same  nwnher  does  not  cha7ige  its  value. 

82.  Rules. — To  reduce  a  fraction  to  its  lowest  terms, 
1.  Divide  both  temns  of  the  fraction  by  any  common  divisor; 
then  divide  both  terms  of  the  resulting  fraction  by  any  common 
divisor ;  and  so  on,  until  the  terms  of  the  resulting  fraction  have 
no  common  divisor  except  1.     Or, 

2.  Divide  both  terms  of  the  fraction  by  ihdr  greatest  common 
divisor. 

Case   IV. 

Fractioias  reduced  to  Higlier  Tex-ms,  and  to  a  Coin- 
ixioii  IDenoininator. 

66.  How  many  eighths  of  a  foot  in  1  fourth  of  a  foot?  In 
2  fourths  ?   3  fourths  ? 

67.  How  many  twelfths  in  3  sixths  ?   4  sixths  ?   5  sixths  '! 

68.  How  many  fifteenths  in  |?   |?  |? 

69.  Change  |,  f ,  and  ^  each  to  twelfths. 

70.  Change  f ,  f ,  and  /^  to  fortieths. 

71.  Change  f,  y%,  and  -j^^  to  sixtieths. 

72.  Change  f ,  -f ,  and  j\-  to  thirtieths. 

73.  Change  f,  f,  and  ^^  to  fortieths. 

74.  Multiply  both  terms  of  |  by  4,  and  show  that  the 
value  of  the  fraction  is  not  changed. 

75.  Show  that  the  multiplication  of  both  terms  of  anv 
fraction  by  the  same  number  does  not  change  its  value. 

C.Ar.— 5 


50  COiMPLETE  ARITHMETIC. 


WKITTEN  PKOBLEMS. 

76.  Reduce  |-,  |-,  and  ||-  to  equivalent  fractions  having  a 
common  denominator. 

7     1 1*  Reduce  the  fractions  to  twenty-fourths,  thus :  f  = 

^  t       T^  10.7  21.11  22 

10        .2  1        2  2  24  »     8'  1¥)    TI  2  4* 

2f       2i       14 

Reduce  to  a  common  denominator, 

77.  I,     I,    f        80.  -I,  t,  tV  83.  I,    f,  ^,  <i 

78.  I,    I,  t\        81.  i,  f,    I  84.  i  ^\,  \l  ii 

79.  f  A,    ^        82.  i  I,    i  85.  i    f,    I,  /t 

86.  Reduce  |,  ^,  ^\,  and  f^  to  equivalent  fractions  hav- 
ing the  least  common  denominator. 

Process.  The  least  common  multiple  of  8,  16,  24,  and 


60 


90. 

-h>  IT. 

ih 

-]* 

91. 

1,  H. 

i5> 

!!' 

« 

92. 

1,  tt. 

H, 

1S"^J 

Hi 

9ti     II     tf     ft        nominator.      Change    the    fractions   to    96ths. 

5   —   60.       7 42.     11    44.     21. .63 

?■  —  ^5  >     T^  —   96  >     1¥  ^^  »     -^-^  M- 

Reduce  to  the  least  common  denominator, 
88.     I,    I,    I,    i 

kQ     _7         8        1  1       1  H 
^^-     l¥»    yS"'    TTT'    a^ 

DEFINlTIOi\S,  PRINCIPLE,  AND  RULES. 

83.  A  fraction  is  reduced  to  higher  terms  when  it  is 
changed  to  an  equivalent  fraction  with  greater  terms. 

84.  Several  fractions  are  reduced  to  a  Common  Denomi- 
nator when  they  are  changed  to  equivalent  fractions,  with 
the  same  denominator. 

When  the  common  denominator  of  several  fractions  is 
the  smallest  denominator  which  they  can  have  in  common, 
it  is  called  their  Least  Common  Denominator. 

85.  Principle. — TJie  multiplication  of  both  terms  of  a  frac- 
tion by  the.  same  number  does  not  change  its  value. 


REDUCTION  OF  FRACTIONS.  51 

86.  EuLEs. — 1.  To  reduce  a  fraction  to  higher  terms, 
Divide  the  given  denominator  by  the  denominator  of  the  fraction, 
and  multiply  both  terms  by  the  quotient. 

2.  To  reduce  fractions  to  the  least  common  denominator, 
Divide  the  least  common  midtiple  of  the  denominators  by  the  de- 
nominator of  each  fraction,  and  multiply  both  terms  by  the  quotient. 

Case  V. 

CoiTipovind  lETractions  I'ecliaced  to  Siixiple  Fractions. 

93.  How  much  is  1  half  of  1  third  of  a  pear  ?  1  half  of 
1  fourth  of  a  pear? 

94.  A  father  divided  |^  of  a  pine-apple  equally  between  3 
boys :  what  part  of  the  pine-apple  did  each  boy  receive  ? 

95.  What  is  ^  of  i?  \  of  |?   \  of  |  ? 

96.  What  is  ^  of  i?  i  of  f  ? 

97.  What  is  i  of  ^?  i  of  f  ?    I  of  |? 

98.  What  is  I  of  ^?  j  of  I?  f  of  |? 

99.  What  is  I  of  I?  I  of  f  ?   I  of  |? 

100.  What  is  I  of  4?   ^  of  f  ?   I  of  ^? 

101.  What  is  I  of  12?   \  of  12^?   \  of  13^? 

SoLUTiox.— I  of  13|^  =  ^  of  12,  which  is  4,  +  ^  of  1^  or  |,  wliich  is 
f  or  i.     4  +  I  ==  4|.     Hence,  \  of  131  is  ^, 

102.  What  is  \  of  17i?   i  of  21|?   \  of  33^? 

103.  What  is  I  of  12?   I  of  12^?  |  of  16^? 

104.  What  is  f  of  22^?   f  of  25 J?   J  of  37^? 

105.  What  is  g  of  33^?   |  of  42^?  j\  of  62^? 

WKITTEN  PROBLEMS. 

106.  Reduce  %  of  y^  of  7|  to  a  simple  fraction. 

Process. 

f  of  ^  of  7i  =  ^^X-AXl5L225_5    ^,,,, 
^        '"         ^      9  X'15X    2       270      6 

''         ^      0Xi:0X    2       6 
3 


52  COMPLETE  ARITHMETIC. 

Reduce  to  a  simple  fraction, 

107.  I  of  I  of  4  111.  f  of  f  of  f  of  3J 

108.  f  of  ^  of  2^  112.  I  of  y\  of  f  of  4f 

109.  j-  of  If  of  1|  113.  i  of  ^  of  2f  of  2^ 

110.  i  of  f  of  21  114.  I  of  1%  of  tV  of  3i 

87.  Rules. — To  reduce  a  compound  fraction  to  a  simple 
fraction,  1.  Multiply  the  numerators  together  for  a  numerator, 
and  the  denominators  together  for  a  denominator.     Or, 

2.  Indicate  the  continued  multiplication  of  the  numerators, 
and  also  of  the  denominators,  and  reduce  the  resulting  fraction 
to  its  lowest  terms  by  cancellation. 

REVIEW  PROBLEMS. 

115.  Reduce  16  to  a  fraction  having  8  for  a  denominator. 

116.  Change  35-1-21  to  a  fraction  in  its  lowest  terms. 

117.  How  many  15ths  of  a  gallon  in  33^  gallons? 

118.  Reduce  ^/^  to  a  mixed  number  with  the  fraction  in 
its  lowest  terms. 

119.  Reduce  $f g-,  $W,  ^W'  and  $^Y  each  to  whole  or 
mixed  numbers. 

120.  Reduce  |,  /^,  and  \l  each  to  30ths. 

121.  Reduce  12^,  18|,  and  33^  each  to  12ths. 

122.  Reduce  y\  of  ^^  of  2-^  of  24  to  a  simple  fraction. 

123.  Reduce  |,  f ,  and  ^^  to  a  common  denominator ;  to 
the  least  common  denominator. 

124.  Reduce  f,  5^,  and  f  of  -f  to  their  least  coTnmon 
denominator. 

Suggestion. — First  reduce  5|  and  f  of  f  to  simple  fractions. 

125.  Reduce  ^,  J  of  f ,  and  f  of  6|  to  their  least  common 
denominator. 

126.  Reduce  ^  of  2^,  f  of  3,  and  ^  of  ^  of  6^  to  their 
least  common  denominator. 

127.  Reduce  J  of  J,  |  of  2|,  and  -f  of  l^  to  their  least 
common  denominator. 

128.  Reduce  J,  f  of  5^,  2^  of  3J,  and  2y»7  to  their  least 
common  denominator. 


ADDITION  OF  FRACTIONS.  53 


ADDITION   OF  FRACTIONS. 

1.  A  clerk  spends  f  of  his  salary  for  board,  |  of  it  for 
clothing,  and  ^  for  other  expenses:  what  part  of  his  salary 
does  he  spend? 

2.  How  many  ninths  in  -f,  f,  and  ^? 

3.  A  man  traveled  ^  of  his  journey  the  first  day,  and  J 
of  it  the  second  day :  what  part  of  the  journey  did  he  travel 
in  the  two  days? 

4.  How  many  twelfths  in  ^  and  i?    |  and  ^? 

5.  A  owns  I  of  a  vessel,  and  B  |-  of  it :  what  part  of  the 
vessel  do  both  own  ? 

6.  What  is  the  sum  of  |  and  |  ?   |  and  f  ? 

7.  fandyV?         f  and  ^?       |  and    |?     fV  and    I? 

8.  4  and    ^?         ^  and    f?       f  and  ^^^^  ?        |  and    J? 

9.  i  and  51?       2 J  and  6^?     5^  and  6^?      8'  and  9|? 

Suggestion. — First  add  the  fractions  and  then  the  integers. 

10.  Show  that  fractions  having  a  common  denominator, 
express  like  fractional  units,  and  that  only  like  fractional 
units  can  be  added. 

WRITTEN  PROBLEMS. 

11.  What  is  the  sum  of  ^%  ^|,  and  ^? 
Process:     ±  +  il  +  iP  —  9  +  16  +  10  _  3_,5  _  11.2,  Am. 

2  3      '      8  3      '      23—  23  '  2  3  2  3'     ^"^' 

12.  What  is  the  sum  of  \i,  jf,  -]|,  and  yV? 

13.  What  is  the  sum  of  ^,  |?,  ||,  and  f  J? 

14.  What  is  the  sum  of  J-fJ,  j\\,  and  J-gl? 

15.  What  is  the  sum  of  f,  j\,  and  ii? 

Process.   "  Since  unlike  fractional  units  can  not  be  added, 

f  +  1^2.+  xi  =  reduce  the  fractions  f,  f^,  and  \^  to  a  common 

f f  +  4 f  +  If  =  denominator,  and  then   add   the   resulting   frac- 

fi-=r  1||,  Am.  tions. 


54  COMPLETE  ARITHMETIC. 

16.  Add  f  and  If  21.  ^%,  ^%  |3,  and  U- 

17.  I,  ,V,  and  j\.  22.   |,  |,  j^,  and  i^. 

18.  I,  li,  and  J-f.  23.  ||,  if  f ?,  and  f. 

19.  f,  1^0'  and  If.  24.  |,  y^*  A^  and  ^^. 

20.  A   M,  and  fi  25.  ^-L,  _i_,  ^i^,  and  ^^,. 

26.  Add  I,  f  of  f ,  and  f  of  j  of  2-^. 

PJ^ocess.  Since  -|  of  |  =  f ,  and  f  of  | 

of  2^  =  f ,  the  sum  of  |  -f  f  of 

r=lH  f+70ff0f2^=f  +  f  +  i 

27.  Add  I  of  I,  I  of  fV  of  2^,  and  |. 

28.  Add  I  of  2i 'l  of  I,  and  f  of  ^  of  6. 

29.  Add  ^\,  }  of  5,  and  f  of  f  of  |. 

30.  Add  j\  of  2,  I-  of  f ,  i  of  i  of  T%,  and  3f 

31.  Add  33^,  37^,  55f,  and  66|. 

Process. 

ooi^      4  The  sum  of  33|,  37|,  55f,  and  66f,  equals  the  sum 

37 1     Y  of  i  +  ^  +1  +  f  added  to  the  sum  of  33  +  37+  55  +  66. 

553     V  ^   i  +  ^  +  f  +  t==2x\or2f    Write  the  i  under  the  frac- 

n(^2      8  tions  and  add  the  2  with  the  integers.     The  sum  is 


193^,  ^ns. 


32.  Add  39i,  56f,  88|-,  and  104^^^. 

33.  Add  45,  87f,  66|,  and  75i. 

34.  Add  121,  16|,  18f,  30f  33i,  and  62^. 

35.  Add  f ,  I  of  f ,  16|,  and  48^. 

36.  Add  $5.12^,  $3.18|,  $8.25,  and  $3.81j. 

37.  Add  f,  j%  }  of  5i,  and  65/x. 

38.  Add  f ,  I  l^\,  and  ,\  of  2|. 

PRINCIPLES  AND  RULES. 

88.  Principles. — 1.  Only  like  fractional  units  can  be  added. 
Hence, 

2.  Fractions  must  have  a  common  denominator  before  they  can 
be  added. 


SUBTRACTION  OF  FRACTIONS.  00 

89.  Rules. — 1.  To  add  fractions,  Reduce  the  fractions  to  a 
common  denominator,  add  the  numerators  of  the  nmu  fractions, 
and  under  Hie  sum  write  tJie  common  denominator. 

2.  To  add  mixed  numbers,  Add  the  fractions  and  the  in- 
tegers separately,  and  combine  the  results. 

Notes. — 1.  Compound  fractions  must  be  reduced  to  simple  fractions 
before  they  can  be  added. 

2.  When  mixed  numbers  are  small  they  may  be  reduced  to  im- 
proper fractions  and  then  added. 

SUBTRACTION  OF  FRACTIONS. 

1.  A  boy  spent  |  of  his  money  for  a  slate:  what  part  of 
his  money  has  he  left? 

2.  How  much  is  f  less  f  ?     f  less  f?     f  less  f? 

3.  How  much  is  \^  less  3^?     ]-^  less  /j?     |^  less  j\? 

4.  A  bought  f  of  a  bushel  of  clover  seed  and  sold  ^  of  a 
bushel  to  B :  what  part  of  a  bushel  has  A  left  ? 

Suggestion.— Change  f  and  ^  to  twelfths. 

5.  How  much  is  f  less  |  ?     J  less  |  ?     |  less  J? 

6.  f  less  f  ?     f  less  f  ?     |  less  |?     |  less  J  ? 

7.  ^  less  f  ?    I  less  ^?    f  less  |  ?     f  less  |? 

8.  f  less  I-?     I  less  f  ?     |  less  |?     ^  less  |? 

9.  1^^  less  I?     fi  less  |?     |  less  jV  •     I  less  |  ? 

10.  51  less  3|?     6|  less  4^?     9  J  less  7^?     12^  less  GJ? 

11.  Why  can  not  |  be  subtracted  from  |  without  first  re- 
ducing the  fractions  to  a  common  denominator? 

WRITTEN  PROBLEMS. 

12.  Subtract  i|  from  ||. 

Pkocess:     ||-i|  =  27-i9_^^^^^ 

13.  Subtract  |i  from  {§ ;   ^  from  |f;  f|  from  ff 

14.  Subtract  ^2^  from  jf 

Process  :    \i  -  /^  =  M  -  M  =  M,  ^ns. 


56  COMPLETE  ARITHMETIC. 


How 

much  is 

15. 

a-  f? 

18. 

il-U? 

21. 

M-A? 

16. 

H-A? 

19. 

M-ii? 

22. 

2  3  17  ? 

3«           54   • 

17. 

^-vV? 

20. 

A-A? 

23. 

It-i*? 

24.  From  |  of  4  take  |^  of  J  of  |. 

Process  :     \  of  *  -  |      f  of  |  of  |  —  |     f  —  -^  =  ^%,  ^?is. 

25.  From  |  of  f  take  |  of  J  of  |. 

26.  From  |  of  7  take  i  of  |  of  7. 

27.  From  ^  of  |  of  |  take  f  of  |  of  f . 

28.  From  |  of  ^  of  2^  take  j\. 

29.  From  340|  take  247f . 

Process.  First  subtract  the  fractions  and  then  the  integers. 

0-Q2        8  Since  h%  is  greater  than  -2%,  add  | g-  to  ^^0,  making  j^, 

^^  and  then  subtract  ^f  from   y,  writing  the  diiference, 

4'       2  0  i.|^  under  the  fractions,  and  adding  1  (|^)  to  the  7 

92|-|,  -4ws.  units  before  subtracting  the  integers. 

30.  93|  — 46^  =  ? 

31.  561  —  37^:-=? 

32.  108f  — 901=:? 

36.  What  fraction  added  to 

37.  What  number  added  to  6f  will  make  16 f  ? 

38.  From  the  sum  of  §  and  |  take  their  difference. 

39.  From  f-f  |  take  |  — |. 

40.  From|  +  |+T^o   take  ^  of  If 

41.  From  -J  +  I  take  -^-^ —  ^  ^f  5^ 

42.  From  a  cask  containing  45^  gallons  of  sirup,  a  grocer 
sold  one  customer  16|  gallons  and  another  21f  gallons:  how 
many  gallons  remained  unsold  ? 

43.  A  man  bequeathed  yu  ^^  ^^^  property  to  his  wife,  j\ 
of  it  to  his  children,  and  the  remainder  to  a  college  for  its 
better  endowment.  What  part  of  his  property  did  the  col- 
lege receive  ? 

44.  A  man  owning  f  of  a  factory,  sold  ?  of  his  share : 
what  part  of  the  factory  did  he  still  <nvii  ? 


33. 

241 J - 

153/^:=? 

34. 

$2. 33 J  - 

-$1.62|r. 

-? 

35. 

$3,121- 

-  $2.48  J- 

-? 

1  ^ 

nil  make 

H? 

MULTIPLICATION  OF  FRACTIONS.  57 

45.  Two  ninths  of  a  pole  is  in  the  mud,  f  of  it  in  the 
water,  and  the  rest  of  it  in  the  air :  what  part  of  the  pole  is 
in  the  air  ? 

46.  The  part  of  a  pole  broken  off  by  the  wdnd  was  f  of 
the  whole  pole,  and  f  of  the  part  still  standing  was  above 
the  ground:  what  part  of  the  pole  was  in  the  ground? 

PRINCirLES  AND  RULES. 

90.  Principles. — 1.  The  mmwend  and  subtraliend  must  de- 
note like  fractional  units.     Hence, 

2.  Fractions  must  have  a  common  denominator  before  their 
difference  can  be  found. 

91.  Rules. — 1.  To  subtract  fractions.  Reduce  ihe  fractions 
to  a  common  denominatorr,  subtract  the  numerator  of  the  subtra- 
hend from  the  numerator  of  the  minuend,  and  under  the  differ- 
ence write  the  common  denominator. 

2.  To  subtract  mixed  numbers,  Subtract  first  the  fractions, 
and  then  the  integei's,  and  unite  the  residts. 

Notes. — 1.  Compound  fractions  must  be  reduced  to  simple  fractions 
before  they  can  be  subtracted. 

2.  AVhen  mixed  numbers  are  small  they  may  be  reduced  to  im- 
proper fractions,  and  then  subtracted. 

MULTIPLICATION  OF  FRACTIONS. 
Case  I. 

Fractions    multiplied  Toy   Integers. 

1.  How  much  is  twice  2  ninths  of  an  inch?  .  4  times  2 
ninths  of  an  inch? 

2.  If  a  basket  hold  f  of  a  bushel,  how  many  bushels  will 
8  baskets  hold  ?     10  baskets  ? 

3.  How  much  is  8  times  f  ?     10  times  J?     20  times  f  ? 

4.  6  times  ^?     8  times  J  ?     9  times  yV  •      12  times  |  ? 

5.  7  times  j^  ?     9  times  j\  ?    8  times  ^J  ?     11  times  |  ? 

6.  6  times  5^?     9  times  6f?     7  times  12^?     10  times  7|? 

7.  8  times  12|?  6  times  16|?  5  times  33^?  7  times 
30i? 


58  COMPLETE    ARITHMETIC. 

8.  Why   does   multiplying   the   numerator  of  -j^  by  3 
multiply  the  fraction  by  3? 

9.  Why  does  dividing  the  denominator  of  ^  by  3  mul- 
tiply the  fraction  by  3  ? 

10.  In  how  many  ways  may  a  fraction  be  multiplied  by 
an  integer? 

ATITRITTEW  PKOBLEMS. 

Multiply 

11.  -^  by  9.  15.  ^\  by  12.  19.  62^  by  36. 

12.  ^f  by  10.  16.  ^\\  by  16.  20.  45f  by  80. 

13.  A  by  24.  17.  ^  by  60.  21.  $5.18f  by32. 

14.  H  by  45.  18.  m  by  25.  22.  $661  by  52. 

PRINCIPLE  AND  RULES. 

92.  Principle. — A  fraction  is  multiplied  hy  multiplying  its 
nwnerator  or  dividing  its  denominator. 

93.  Rules. — 1.  To  multiply  a   fraction   by  an   integer, 
Multiply  the  numerator  or  divide  the  denominator. 

2.  To  multiply  a  mixed  number  by  an  integer,  Midtiply 
Hie  fraction  and  the  integer  separately,  and  add  the  products. 

Case  II. 

Integers   multiplied  "by  Fractions. 

23.  If  a  ton  of  hay  cost  $16,  what  will  ^  of  a  ton  cost? 
J  of  a  ton  ? 

24.  If  an  acre  of  land  is  worth  $50,  what  is  ^  an  acre 
worth?     I  of  an  acre? 

25.  What  is  I  of  42?  f  of  42?  i  of  42? 

26.  What  is  f  of  56  ?  |  of  56?   |  of  56? 

27.  I  of  63?  -^  of  84?  ^  of  99?  |  of  56? 

Solution.—  ^  of  56  =  6f ,  and  |  of  56  =  7  times  6|  =  43^. 

28.  f  of  66?  i  of  66?  ^  of  66?   f  of  74? 


Process. 

654 

54^ 

7 

Or: 

38H, 

Ans, 

MULTIPLICATION  OF  FRACTIONS.  59 

29.  AVhat  is  16  X  f  ?   50  X  |?   42  X  |? 

Solution.— Since  f  =  f  of  1,  16  X  |=  f  of  16  X  1  ^  I  of  16  =  12. 

30.  57X1?   75X1?   87  X  A?  95  X  A?     76  Xf? 

31.  47  X  I?  68  X  J?   75  X    f?   83Xiftr?   100  X  A? 

32.  Show  that  the  product  of  an  integer  by  a  fraction 
equals  the  fraction  of  the  integer. 

WRITTEN  PROBLEMS. 

33.  Multiply  654  by  ■^. 

Process. 

12  )  654  654  Since  yV  =  7  times  j^j,  or  yV  of  7, 

L_  the  product  of  654  X  tV  =  ^  times 

12  )  4578  ^1^  of  654,  or  j\  of  7  times  654. 

3811  "     • 

34.  66  by  f         37.  784  by  if.       40.  757  by  |  of  f . 

35.  dS  by  if.       38.  648  by  ||.       41.  908  by  J  of  2^. 

36.  92  by  2%.       39.  564  by  |f       42.  588  by  -3^  of  3^. 
43.  Multiply  256  by  27|.     406  by  33f . 

Suggestion.— Since  256  X  27f  =  256  X  27  +  256  X  f,  first  mul- 
tiply by  the  integer  and  then  by  the  fraction,  and  add  the  products. 

44.  66  by  8|.   47.  645  by  12f.  50.  745  by  60f . 

45.  72  by  9f.   48.  465  by  18|.  51.  385  by  45tV 

46.  96  by  8i=V.  49.  406  by  33^.  52,  708  by  60f . 

PRINCIPLE  AND  RULES. 

94,  Principle. — The  product  of  an  integer  by  a  fraction 
equals  tJie  fraction  of  Uie  integer. 

95.  Rules. — 1.  To  multiply  an  integer  by  a  fraction, 
(1)  Divide  the  integer  by  the  denominator,  and  multiply  the 
quotient  by  the  numerator.  Or,  (2)  Multiply  the  integer  by  the 
numerator,  and  divide  the  jyroduct  by  the  denominator. 

2.  To  multiply  an  integer  by  a  mixed  number,  Multiply 
by  the  integer  and  the  fraction  separately,  and  add  the  products. 


60  COMPLETE  ARITHMETIC. 

Case  III. 

Fractions   aniiltiplied  by   Fractions. 

53.  What  is  i  of  i?   I  of  I?   I  of  f  ? 

54.  I- of  f?   I  of  I?   I  of  I?   f  of  I-? 

55.  What  is  fX  I?   fXf?   iX^?   i^XA? 
Suggestion.—  f  X  f  =  f  of  |  ;  -|  X  f  =  |  of  f ,  etc. 

56.  IXt^j-?   f  XH?   AX-f?   IXH?   iXA? 

57.  What  is  i  of  121?   |  of  13J?   |  of  13^. 

Solution.—  ^  of  13^  -=  i  of  12  +  |  of  1^  =  4  +  xi  ==  ^^^ ;  and  | 
of  13^  =  2  times  4x^2  =  8|. 

58.  I  of  16|?  I  of  221?  I  of  421?  ^  of  62^? 

59.  I  of  371?  I  of  42|?  |  of  65^?  -j%  of  lOOi? 

60.  Show  that  |  x  f  =  f  of  |. 

WRITTEN  PROBLEMS. 

61.  Multiply  if  by  f. 

Process  ^^^^^  I  ^  i  of  3,  the  product  of 

•  ||Xf=:i  of   3  times   H  -  i  of 

HX  |  =  7f$i  =  M,  Ans.  13  X  3  ^  13  X  3  _  ,3 
15X4  15  15X4""-^'^' 

62.  H  by  H.  66.  f  by  |  of  |.  70.  2^  by  3J. 

63.  1^  by  1^.  67.  i  of  A  by  -,%  71.  4J  by  5^. 

64.  K  by  H-  68.  i  by  f  by  |.  72.  6^  by  3|. 

65.  H  by  ||.  69.  2|  by  -ft  of  if.  73.  lOi  by  2|. 

74.  What  will  J  of  a  yard  of  cloth  cost  at  $|-  a  yard? 
At  $^  a  yard? 

75.  What  will   5^  pounds  of  flour   cost  at  4^  cents   a 
pound  ?     At  6^  cents  a  pound  ? 

76.  What  will  2|  pounds  of  tea  cost  at  $lf  a  pound? 
At  $li  a  pound  ? 

77.  AVhat  is  the  cost  of  35  barrels  of  flour  at  $6^  a 
barrel  ?     At  $7^  a  barrel  ? 


DIVISION  OF  FRACTIONS.  61 

78.  A  man  owned  -^  of  a  ship  which  w^as  sold  for  $13250 : 
what  was  his  share  of  the  money? 

79.  What  is  the  product  of  ^»  t  of  2^,  |  of  yV  of  ff, 
and  2i  ? 

80.  What  will  V2\  pounds  of  butter  cost  at  18f  cents  a 
pound  ?     At  22^  cents  a  pound  ? 

PRINCIPLE  AND  RULES. 

96.  Principle. — Tlie  product  of  a  fraction  by  a  fraction 
equals  the  frojction  of  the  fraction. 

97.  Rules. — 1.  To  multiply  a  fraction  by  a  fraction, 
Multiply  tJie  numerators  together,  and- also  the  denominators. 

2.  To  multiply  a  mixed  number  by  a  mixed  number, 
Reduce  the  mixed  numbers  to  improper  fractions^  and  proceed 
as  above. 

Notes.— 1.  Mixed  numbers  may  be  muhiplied  together  by  first  mul- 
tiplying the  integers  ;  next  multiplying  each  integer  by  the  fraction  united 
with  the  other  integer ;  next  multiplying  the  two  fractions  /  and  then  add- 
ing the  four  products.  Thus,  18f  X  12 J  =  18  X  12  +  18  X  ^  -f  12  X 
I  -f  I  X  i-  But  in  most  cases  it  is  shorter  to  reduce  the  mixed 
numbers  to  improper  fractions. 

2.  Cases  I  and  II  may  be  included  in  Case  III,  by  changing  the 
integer  to  the  form  of  a  fraction.  Thus,  |  X  5  =  -f  X  f ,  and  8  X  f  = 
fXf. 

3.  The  process  of  multiplying  fractions  may  be  shortened  by  can- 
cellation. Compound  fractions  need  not  be  reduced  to  simple  frac- 
tions, since  I  X  I  of  i^  =  I X  f  X  I?- 

DIVISION   OF  FRACTIONS. 
Case  I. 

I^ractiorLS    divitled   by   Integers. 

1.  If  a  man  can  do  y  of  a  piece  of  work  in  3  days,  how 
much  can  he  do  in  1  day? 

2.  A  man  divided  f  of  a  farm  equally  between  4  sons  : 
what  part  of  the  farm  did  each  receive? 

3.  If  5  yards  of  muslin  cost  4  of  a  dollar,  what  will  1 
vard  co?t  ? 


62  COMPLETE  ARITHMETIC. 

4.  If  10  oranges  cost  f  of  a  dollar,  what  will  1  orange 
cost? 

5.  If  8  bushels  of  oats  cost  $2f ,  what  will  1  bushel  cost  ? 

6.  If  fj  of  a  melon  be  divided  into  5  equal  parts,  what 
will  each  part  be? 

7.  Why  does  dividing  the  numerator  of  f  by  4  divide 
the  fraction  by  4? 

8.  Why  does  multiplying  the  denominator  of  f  by  4 
divide  the  fraction  by  4? 

9.  In  how  many  ways  may  a  fraction  be  divided  by  an 
integer  ? 

WRITTEN   PROBLEMS. 

10.  Divide  ^  by  6. 


Since  H  - 

-i  =  M,  H^6  =  iof 

Process. 
1 2  -^  fi 

12       12-i-6 
'"^          25    ' 

12 
or  — -.    Or,  since  to 

25X6 

H^«==      25      =*' 

Ans. 

divide  a  number  by  6  is  to  find  ^  of 

*^-    «^«  =  25X6^ 

=  ^V 

of  it,    M-^6: 

12 

25  X  6* 

=  iofU  =  '^^^or 

Divide 

11.  If  by  8. 

14. 

If  by  15. 

17.  2^  by  8. 

12.  a  by  7. 

15. 

M  by  20. 

18.  5i  by  12. 

13.  If  by  11. 

16. 

n  by  25. 

19.  6f  by  10. 

PRINCIPLE  AND  RULES. 

98.  Principle. — A  fraction  is  divided  by  dividing  its  nu- 
merator or  multiplying  its  denominator. 

99.  Rules. — 1.  To  divide  a  fraction  by  an  integer.  Di- 
vide the  numerator  or  multiply  the  denominator. 

2.  To  divide  a  mixed  number  by  an  integer,  (1)  Reduce 
the  mixed  number  to  an  improper  fraction  and  divide  as  above ; 
or,  (2)  Divide  tlie  integral  part  and  then  the  fraction,  and  unite 
the  quotients. 


DIVISION  OF  FRACTIONS.  63 

Case   II. 

Integers    divided  \:>y  I^ractions. 

20.  How  many  times  is  |  of  a  cent  contained  in  4  cents  ? 

Solution. — In  four  cents  there  are  20  fifths  of  a  cent,  and  2  fifths 
of  a  cent  are  contained  in  20  fifths  of  a  cent  10  times. 

21.  If  a  fruit  jar  hold  f  of  a  gallon,  how  many  jars  will 
hold  6  gallons?     12  gallons?     18  gallons? 

22.  If  f  of  a  yard  of  silk  will  make  a  vest,  how  many 
vests  will  5  yards  make?     7  yards?     10  yards? 

23.  If  a  yard  of  cloth  cost  $f,  how  many  yards  can  be 
bought  for  $10  ?     For  $15  ?     For  $20  ? 

24.  How  many  times  is  |  contained  in  8?  f  in  12?  f  in 
9  ?   I  in  15  ?   I  in  9  ?   f  in  12  ? 

25.  How  many  times  is  f  contained  in  12?^  in  15? 

26.  Show  that  8  --  f  ==  ^^. 

o 

WRITTEN   PROBLEMS. 

27.  What  is  the  quotient  of  25  -f-|? 

Process  :    25  -:-  |  =-  ^^^  ^  28f ,  Ans. 

Note. — It  will  be  noticed  that  the  integer  is  muhiplied  by  the 
denominator  of  the  fraction  and  the  product  divided  by  its  nu- 
merator. 


What  is  the  quotient  of 

28.  21 --3^?        31.  100--!^? 

34.     75  - 

-6i? 

29.  42 --14?        32.     96 --tt? 

35.  120- 

-3i? 

30.  72  ^|4?        33.  125  ^li? 

36.  225- 

-5i? 

100.  Rules. — To  divide  an  integer  by  a  fraction,  1.  Mul- 
tiply the  integer  by  the  denominator  of  the  fraction,  and  divide 
the  product  by  the  numerator.     Or, 

2.  Divide  the  integer  by  the  numerator,  and  multiply  the  quo- 
tient by  the  denominator. 


64  COMPLETE  ARITHMETIC. 

Case  III. 

Fractions   clivitled  by  Fractions. 

37.  How  many  times  is  f  of  an  inch  contained  in  i  of  an 
inch  ?     -I  of  an  inch  in  4  of  an  inch  ? 

38.  How  many  times  f  in  f  ?   |  in  f  ?   f  in  Y? 

39.  How  many  times  |  in  |?    |  in  V?    f  in  Y-?    f  in 

15?     _6_   ;„   _9_9     _6_  in    13  9 
T  •      1 1    "M  1  •      1 1    "*    1 1  • 

40.  How  many  times  is  ^  contained  in  f  ?   -^  in  |? 
Suggestion. — Change  the  fractions  to  tAvelfths. 

41.  How  many  times  |  in  |?    f  in  |-  ?   i  in  |?    f  i^i  A? 

49        3     in    4  ?      3    in      7    ?       3     in     1  1  *?      2    in    3  9      2    in    5  9 

43.  Show  that  the  quotient  of  two  fractions  having  a  com- 
mon denominator,  equals  the  quotient  of  their  numerators. 

-WKITTEN  PKOBLEMS. 

44.  Divide  |  by  |. 

Peocess  :     |.  -  f  =  |^-|  =  H  =  IH,  Ans. 

Since  |  =  ^><^,  and  f=.^><-8,  |_^|^7X_5  ^3X8^7^6^ 
^         40    '  ^         40    '  ^      ^         40  40         3X8 

It  is  thus  seen  that  inverting  the  terms  of  the  divisor,  and  taking  the 
product  of  the  numerators  for  the  numerator,  and  the  product  of  the 
denominators  for  the  denominator,  is  the  same  as  reducing  the  frac- 
tions to  a  common  denominator,  and  dividing  the  numerator  of  the 
dividend  by  the  numerator  of  the  divisor. 

7  V  5 
Note.— That  |  -^  f  =     ^      mav  also  be  thus  explained :    ^  =  3 

times  i  and  since  1  ~  4  =r.  1-XA    t  --  ^  =  \  of  ^^^  =  ^^. 

What  is  the  quotient  of 

45.  A^ii?      49.     3i-2i?      53.  A-^4off|? 

46.  i|-^-TV?      50.     5i-3i?      54.     |ofA-|of4? 


47.  A 

48.  H-^A-      52.  16^ 


il?     51.     6-1-121-?    55.  -jV of  31- 1 of  2i? 
o  o       ra    ..,.    .   31^9      56_  ||^^of|of3-|? 


COMMON   DIVISOR.  65 

57.  If  a  family  use  ^  of  a  barrel  of  flour  in  a  month, 
how  long  will  2^  barrels  last  ? 

58.  If  a  bushel  of  corn  cost  $|,  how  many  bushels  can 
be  bought  for  $^?     For  $9 J? 

59.  If  13  yards  of  silk  cost  $171   how  many  yards  can 
be  bought  for  $48f?     For  $62^? 

60.  If  a  man  walk  3^  miles  an  hour,  in  how  many 
hours  will  he  walk  20J  miles  ? 

61.  At  $33^  an  acre,  how  many  acres  of  land  can  be 
bought  for  $841|? 

62.  By  what  must  f  be  multiplied  that  the  product  may 
be  26|? 

63.  Divide  the  product  of  6^  multiplied  by  3^  by  the 
quotient  of  4^  -4-  5^  ? 

PRINCIPLES  AND  RULES. 

101.  Principles. — 1.  The  quotient  of  two  fractions  having 
a  common  denominator,  equals  the  quotient  of  their  numerators. 

2.  The  multiplying  of  both  dividend  and  divisor  by  the  same 
number  does  not  change  the  value  of  the  quotient. 

102.  Rules. — To  divide  a  fraction  by  a  fraction,  1.  Re- 
duce the  fractions  to  a  common  denominator,  and  divide  the 
numerator  of  the  dividend  by  the  numerator  of  the  divisor.    Or, 

2.  Invert  the  terms  of  the  divisor,  and  then  midtiply  the 
numerators  together  and  also  the  denominators.     Or, 

3.  Midtiply  both  dividend  and  divisor  by  tJie  least  common 
multiple  of  the  denominators  of  the  fractions,  and  divide  the 
resulting  dividend  by  the  resulting  divisor. 

Notes. — 1.  The  third  rule  depends  on  the  second  principle;  and, 
since  multiplying  two  fractions  by  their  least  common  multiple 
changes  them  to  integers,  the  new  dividend  and  divisor  are  always 
integral.  Thus,  multiplying  both  fractions  by  24,  tlie  /.  c.  m., 
f-f- y'j -^  15 -^- 14=  lj\f ;  multiplying  by  6,  the  /.  c.  m.,  6-| -f- 5^  = 
40  -T-  33  —  1^^.  Compound  fractions  should  first  be  reduced  to  simple 
fractions. 

2.  It  is  Hot  necessary  that  the  pupil  be  made  equally  familiar  with 
these  three  methods  of  dividing  one  fraction  by  another.    He  should 
thoroughly  master  one  of  them. 
C.Ar.—C) 


66  COMPLETE  ARITHMETIC. 


COMPLEX  FEACTIONS. 


4 

64.  Reduce  the  complex  fraction  -I,  to  its  simplest  form. 


Process 

■   1 

^i-f  = 

-4X7^ 
5X6 

H,  Ans. 

3  to  the  simplest 

form 

1-         69. 

16| 
25 

73. 

-foff 
foff 

77. 

f-A 

1         70. 
24 

25 
16| 

74. 

?of2i 
5* 

78. 

f  +  f 

L^        71. 
* 

12* 

1 

75. 

79. 

s  - 

1 
12* 

76. 

toftt 

80. 

1  .  1 
1  ■  1 

65. 
66. 
67. 
68. 


103.  A  complex  fraction  expresses  an  unexecuted  divis- 
ion, the  numerator  being  the  dividend  and  the  denomi- 
nator the  divisor.  It  is  reduced  to  its  simplest  form  by 
performing  the  division  as  expressed. 

Notes. — 1.  A  complex  fraction  may  be  changed  to  a  fraction  with 
integral  terms,  by  multiplying  both  of  its  terms  by  the  least  common  mul- 
tiple of  the  denominators  of  its  fractions.  (Art.  102,  Note  1.)  Compound 
fractions  must  first  be  reduced  to  simple  fractions. 

2.  Let  the  above  problems  also  be  solved  by  this  method. 


NUMBERS  PARTS  OF  OTHER  NUMBERS. 

MENTAL  PKOBLEMS. 

1.  If  ^  of  a  barrel  of  flour  cost  $3,  what  will  a  barrel 
cost  ? 

2.  If  ^  of  a   ream   of  note   paper  cost   75  cents,  what 
will  a  ream  cost  ? 

3.  Charles  gave  Henry  7  marbles,  which  were  ^  of  all 
he  had :  how  many  marbles  had  Charles  ? 


DIVISION.  67 

4.  15  is  I  of  what  number? 

5.  16  is  -^  of  what  number? 

6.  12J  is  -|-  of  what  number? 

7.  16f  is  ^  of  what  number? 

8.  22|  is  i  of  what  number? 

9.  24  is  f  of  what  number? 

Solution. — ^^If  24  is  f  of  a  number,  A  is  i  of  24,  which  is  12.  If 
12  is  ^  of  a  number,  |  is  5  times  12,  or  60.     Hence,  24  is  |  of  60. 

10.  27  is  f  of  what  number? 

11.  45  is  1^  of  what  number? 

12.  64  is  -jSj-  of  what  number? 

13.  27^  is  f  of  what  number? 

14.  46f  is  -j^  of  what  number? 

15.  37^  is  f  of  what  number? 

16.  87^  is  -^j  of  what  number? 

17.  45  is  ^  of  how  many  times  9  ? 

18.  63  is  -J  of  how  many  times  12? 

19.  81  is  ^  of  how  many  times  20? 

20.  108  is  H  of  how  many  times  15  ? 

21.  What  part  of  4  is  1  ?     What  part  of  4  is  3? 

22.  What  part  of  6  is  5  ?   9  is  8  ?   12  is  6  ? 

23.  11  is  7?   16  is  12?   20  is  15?   18  is  12?   30  is  15? 

24.  7  is  what  part  of  21?   8  of  32?   9  of  27? 

25.  13  of  39  ?   16  of  72  ?   15  of  25?   60  of  90? 

26.  i  is  what  part  of  f  ?  i  of  |?  ^  of  f  ?  f  of  J? 

27.  iof  i?   I  of  I?   f  of  I?   I  of  A?   |of  li? 

28.  I  of  11?  f  of  4?  I  of  10?   4  of  8?   j-  of  10? 

29.  5i  of  16|?   6J  of  33|?    m  of  37^?   33|  of  16|? 

30.  3|  of  6^?   5^  of  2|?   21  of  3^?    6\  of  12^? 

PRINCIPLE  AND  RULE. 

104.  Principle. — Only  like  numbers  can  be  compared. 

105.  Rule.  — To  find  what  part  one  number  is  of  another, 
Divide  the  number  denoting  the  part  by  the  number  denoting 
the  whole. 


bg  COMPLETE  ARITHMETIC. 

KEVIEW   OF   FRACTIONS. 

MENTAL   PROBLEMS. 

1.  A  boy  having  $J  gave  S|  for  a  knife :  how  much  money 
had  he  left? 

2.  If  f  be  added  to  a  certain  fraction,  the  sum  will  be  ^ : 
what  is  the  fraction? 

3.  A  laborer  spends  f  of  his  wages  for  board  and  ^  for 
clothing:  what  part  has  he  left? 

4.  A  man  did  -^  of  a  piece  of  work  the  first  day,  ^  of  it 
the  second  day,  ^  of  it  the  third  day,  and  the  remainder  the 
fourth  day:  what  part  of  the  work  did  he  do  the  fourth  day? 

5.  A  man  bought  a  farm,  paying  f  of  the  price  down,  ^ 
of  it  the  first  year,  ^  the  second  year,  and  the  remainder 
the  third  year:  what  part  did  he  pay  the  third  year? 

6.  A  man  is  42  years  of  age,  and  ^  of  his  age  equals  the 
age  of  his  son :  how  old  is  the  son  ? 

7.  A  man  bought  a  cow  for  $33^  and  sold  her  for  |  of 
what  she  cost :  how  much  did  he  lose  ? 

8.  If  a  yard  of  velvet  cost  $8^,  what  will  f  of  a  yard 
cost? 

9.  Jane's  age  is  16|  years,  and  Mary's  age  is  f  of  Jane's : 
how  old  is  Mary? 

10.  A  man  owning  -f  of  a  mill  sells  f  of  his  share:  what 
part  of  the  mill  does  he  still  own? 

11.  Charles  bought  f  of  a  pound  of  candy  and  gave  his 
sister  f  of  a  pound,  and  his  playmate  f  of  what  remained: 
what  part  of  a  pound  had  he  left? 

12.  A  wife  is  35  years  of  age,  and  her  age  is  4  of  the  age 
of  her  husband:  how  old  is  her  husband? 

13.  The  diflference  between  f  and  f  of  a  certain  number 
is  14:  what  is  the  number? 

14.  A  farmer  sold  50  sheep,  which  were  f  of  his  flock: 
how  many  sheep  had  he  before  the  sale? 

15.  When  Charles  is  -f  older  than  he  now  is,  he  will  be 
21  years  of  age:  how  old  is  he? 


REVIEW  PROBLEMS.  69 

16.  A  farmer  sold  f  of  his  farm  for  $1645:  at  this  rate, 
■what  was  the  value  of  the  farm  ? 

17.  A  man  sold  f  of  his  farm  and  had  64  acres  left: 
how  many  acres  had  he  at  first  ? 

18.  A  man  sold  a  horse  for  $90,  which  was  }  more  than 
it  cost  him :  what'  was  the  cost  of  the  horse  ? 

19.  A  lady  paid  $30  for  a  cloak,  which  was  f  more  than 
she  paid  for  a  dress :  what  was  the  cost  of  the  dress  ? 

20.  f  of  42  is  ^  of  what  number  ? 

21.  A  man  is  45  years  old,  and  |  of  his  age  is  f  of  the 
age  of  his  wife:  how  old  is  his  wife? 

22.  Samuel  is  |  as  old  as  Harry,  and  James,  who  is  9 
years  old,  is  f  as  old  as  Charles:  how  old  are  Charles  and 
Samuel  ? 

23.  A  man  gave  $150  for  a  watch  and  chain,  and  the 
chain  cost  f  as  much  as  the  watch :  what  did  each  cost  ? 

24.  If  to  A's  age  there  be  added  |  and  f  of  his  age,  the 
sum  will  be  48  years :  what  is  A's  age  ? 

25.  A  farmer's  sheep  are  in  4  fields;  the  first  contains  f 
of  all,  the  second  ^,  the  third  |,  and  the  fourth  46  sheep : 
how  many  sheep  in  the  4  fields? 

26.  A  saddle  cost  $35,  and  -f  of  the  cost  of  the  saddle 
was  f  of  the  cost  of  a  bridle :  what  was  the  cost  of  the 
bridle? 

27.  If  to  f  of  a  man's  age  15  years  be  added,  the  sum 
will  be  f  of  his  age :  how  old  is  he  ? 

28.  The  distance  from  Cleveland  to  Columbus  is  138 
miles,  f  of  which  is  |^  of  the  distance  from  Columbus  to 
Cincinnati :  what  is  the  distance  from  Columbus  to  Cin- 
cinnati ? 

29.  f  is  f  of  what  number? 

30.  If  I  of  the  value  of  a  house  equals  f  of  the  value  of 
a  lot,  and  the  value  of  both  is  $4400,  what  is  the  value  of 
each? 

31.  If  -|  of  A's  money  equals  f  of  B's,  and  both  together 
have  $340,  how  much  has  each? 


70  COMPLETE   ARITHMETIC. 

32.  If  I  of  A's  age  is  f  of  B's,  and  |  of  B's  is  20  years : 
what  is  the  age  of  each  ? 

33.  If  ^  of  a  yard  of  velvet  cost  $2|,  what  will  |  of  a 
yard  cost? 

34.  How  many  pounds  of  honey,  at  $|  a  pound,  can  be 
bought  for  $3  ?  ^  • 

35.  How  many  bushels  of  apples,  at  $|  a  bushel,  can  be 
bought  for  $16|? 

36.  If  a  barrel  hold  2|  bushels,  how  many  barrels  will 
be  required  to  pack  55  bushels  of  apples? 

37.  If  5|  lb.  of  sugar  cost  $1,  how  much  will  49 1  lb. 
cost? 

38.  If  f  of  a  yard  of  silk  cost  H},  how  many  yards  can 
be  bought  for  $10|? 

39.  If  3f  yards  of  cloth  cost  $5^,  what  will  G^-  yards 
cost? 

40.  If  a  train  of  cars  run  f  of  a  mile  in  If  minutes,  how 
many  miles  will  it  run  in  15  minutes? 

41.  If  4  pounds  of  coffee  cost  $f,  what  will  7-|-  pounds 
cost  ? 

42.  If  12^  tons  of  hay  will  feed  5  horses  a  year,  how 
many  tons  will  feed  8  horses  a  year? 

43.  If  a  rod  5  feet  long  casts  a  shadow  8^  feet  long,  what 
is  the  length  of  a  pole  whose  shadow,  at  the  same  time  of 
day,  is  17|  feet? 

44.  If  3  men  can  do  a  piece  of  work  in  10 J  days,  how 
long  will  it  take  8  men  to  do  it? 

45.  If  a  barrel  of  flour  will  supply  12  persons  4f  weeks, 
how  long  will  it  supply  7  persons  ? 

46.  A  can  do  a  job  of  work  in  12  days,  and  B  in  10  days : 
how  long  will  it  take  both  to  do  it? 

47.  A  and  B  can  do  a  certain  work  in  8  days,  and  A  can 
do  it  in  12  days :  in  what  time  can  B  do  it? 

48.  A  and  B  can  mow  a  field  in  10  days,  and  A  can  mow 
only  f  as  much  as  B :  how  long  would  it  take  each  to  mow 
the  field? 

49.  How  is  the  value  of  a  proper  fraction  affected  by 


REVIEW  PROBLEMS.  7! 

adding  the  same  number  to  both  of  its  terms?     By  sub- 
tracting the  same  number?     (Illustrate,  taking  f.) 

50.  How  is  the  value  of  an  improper  fraction,  greater 
than  1,  aifected  by  adding  the  same  number  to  both  of  its 
terms?     By  subtracting  the  same  number?     (Illustrate.) 

^VRITTEN  PROBLEMS. 

51.  Add  I,  i,  A  of  yV,  and  3i 

52.  From  -|  of  1^  take  4  of  J-|. 

53.  From  the  sum  of  27f  and  20f  take  their  difference. 

54.  Multiply  If  by  35;   35  by  if;  |f  by  f?;   3^  by  21. 

55.  Divide  fj  by  32  ;   32  by  if ;   if  by  |;   4i  by  31. 


9 


56.  H  +  T^-what?  ii-A?  HXtV?  ii-A 

57.  Multiply  2045f  by  35  ;    806  by  84| ;   301  by  16f . 
^  58.  Divide  347f  by  15;   692  by  21|;   19|  by  16f. 

59.  A  farm  is  divided  into  five  fields,  containing  respect- 
ively 21f  A.,  34|  A.,  45 J  A.,  56f  A.,  and  29^  A.:  how 
many  acres  in  the  farm  ? 

60.  There  are  30^  sq.  yd.  in  a  square  rod :  how  many 
square  rods  in  786^  sq.  yd.? 

61.  A  man  travels  5J  miles  an  hour:  how  long  will  it 
take  him  to  make  a  journey  of  75f  miles  ? 

62.  At  $8f  a  ton,  how  many  tons  of  hay  can  be  bought 
for  $1081? 

63.  If  3*3-  of  an  acre  of  land  cost  $68,  what  will  12^  acres 
cost? 

64.  If  f  of  a  yard  of  velvet  cost  $84,  how  many  yards  can 
be  bought  for  $1964? 

65.  If  a  number  be  diminished  by  4  of  4-f  of  itself,  the 
remainder  will  be  69 :  what  is  the  number  ? 

66.  A  pedestrian  walked  -^  of  his  journey  the  first  day, 
f  of  it  the  second  day,  and  then  had  24  miles  to  travel : 
how  long  was  the  journey? 

67.  A  man  pays  $350  a  year  for  house  rent,  which  is  ^4 
of  his  income :  what  is  his  income  ? 

Q8.  A  man  bequeathed  to  his  wife  $4860,  which  was  if 
of  his  estate :  what  was  the  value  of  the  estate  ? 


72  COMPLETE  ARITHMETIC. 

69.  A  graded  school  enrolls  208  boys,  and  -j^  of  the 
pupils  are  girls :  how  many  pupils  are  enrolled  in  the 
school ? 

70.  A  man  owning  f  of  a  ship  sells  f  of  his  share  for 
$3480 :  at  this  rate,  what  is  the  value  of  the  ship  ? 

71.  A  owning  f  of  a  mill,  sold  f  of  his  share  to  B,  and 
^  of  what  he  then  owned  to  C  for  $460 :  what  was  the  value 
of  the  mill  at  the  rate  of  C's  purchase  ? 

72.  A  owns  j^  of  a  section  of  land ;  B,  j^  of  a  section ; 
and  C,  y^^  as  much  as  both  A  and  B  :  what  part  of  a  sec- 
tion does  C  own? 

73.  A  bought  I  of  a  factory  for  $21840,  and  sold  f  of 
his  share  to  B,  and  -|  of  it  to  C  :  what  part  of  the  factory 
did  A  then  own? 

74.  A  and  B  together  own  396  acres  of  land,  and  f  of 
A's  farm  equals  |  of  B's :  how  many  acres  does  each  own  ? 

75.  A  stock  of  goods  is  owned  by  three  partners,  A  own- 
ing f ,  B  1^,  and  C  the  remainder ;  the  goods  were  sold  at 
a  profit  of  $6160:  what  was  each  partner's  share? 

76.  I  of  a  stock  of  goods  was  destroyed  by  fire,  and  f  of 
the  remainder  was  damaged  by  water,  and  the  uninjured 
goods  were  sold  at  cost  for  $5280:  what  was  the  cost  of  the 
entire  stock  of  goods  ? 

77.  A  man  paid  |  of  his  money  for  a  farm,  i  of  what 
remained  for  repairs,  ^  of  what  then  remained  for  stock,  ^ 
of  what  then  remained  for  utensils,  and  then  had  left  $650 : 
how  much  money  had  he  at  first? 

78.  A  merchant  tailor  has  67f  yards  of  cloth,  from  w^hich 
he  wishes  to  cut  an  equal  number  of  coats,  pants,  and  vests : 
how  many  of  each  can  he  cut  if  they  contain  3f ,  2|-,  and 
li  yards  respectively? 

79.  An  estate  was  divided  between  two  brothers  and  a 
sister ;  the  elder  brother  received  |  of  the  estate,  the  younger 
■^jf,  and  the  sister  the  remainder,  which  was  $450  less  than 
the  elder  brother  received:  what  was  the  value  of  the  estate? 
What  was  each  brother's  share  ? 


DECIMALS.  73 

SECTION   IX. 
DECIMAL  FRACTIONS, 

NUMERATION  AND  NOTATION. 

1.  If  a  unit  be  divided  into  ten  equal  parts,  what  is  one 
part  called? 

2.  If  a  tenth  of  a  unit  be  divided  into  ten  equal  parts, 
what  is  one  part  ?     What  is  -^  of  -^  ? 

3.  If  a  hundredth  of  a  unit  be  divided  into  ten  equal 
parts,  w^hat  is  one  part  ?     What  is  -^  of  y^  ? 

4.  What  part  of  a  tenth  is  a  hundredth?  What  part  of 
a  hundredth  is  a  thousandth? 

5.  How  do  the  fractions  ^,  yf g^,  and  y^qq  compare  with 
each  other  in  value?     y^^,  y^,  and  y^u^ 

106.  Since  the  fractional  units,  tenths,  hundredths,  thou- 
sandths, etc.,  decrease  in  value  tenfold,  they  are  expressed, 
like  the  orders  of  integers,  on  a  scale  of  ten.  This  is  done 
by  extending  the  orders  to  the  right  of  units,  and  calling 
the  first  fractional  order  tenths,  the  second  hundredths,  the 
third  thousandths,  etc.  A  period  is  placed  at  the  left  of  the 
order  of  tenths.  Thus,  ^  is  written  .5;  j^  is  written  .05; 
y^"^^  is  written  .005,  etc. 


Copy  and  read 

(6)         (7) 

(8) 

(9) 

(10) 

Cll) 

.4          .03 

.002 

.06 

.07 

.005 

.7          .05 

.004 

.006 

.004 

.4 

.6           .08 

.006 

.08 

.8 

.07 

.9          .09 

.007 

.5 

.09 

.009 

12.  How  many  tenths  and  hundredths  in  .25?     In  .45? 
,63?   .78?   .84?   .69?   .39? 

C.Ar.-7. 


74  COMPLETE  ARITHMETIC. 

13.  How  many  tenths,  hundredths,  and  thousandths  in 
.325?    In  .246?    .307?    .405?    .056? 

14.  How  many  tenths,  hundredths,  and  thousandths  in 
.045?    In  .407?   .008?   .065?   .607?   .325? 

15.  How  many  hundredths  in  yVo?     In  yfo?    -34?    .42? 

16.  How  many  thousandths  in  y||^?  In  y^w?  .325? 
.065?   .205?    .008?   .046? 

107.  When  the  right-hand  figure  of  a  decimal  denotes 
hundredths,  the  whole  decimal  denotes  hundredths,  and 
when  the  right-hand  figure  denotes  thousandths,  the  whole 
decimal  denotes  thousandths.  Thus,  .25  is  read  25  hun- 
dreths:  .325  is  read  325  thousandths. 


Copy  and  read 

(17) 

(18) 

(19) 

(20) 

(21) 

.15 

.016 

.245 

.8 

.007 

.42 

.024 

.354 

.63 

.038 

.36 

.045 

.403 

.086 

.462 

.50 

.083 

.587 

.369 

.507 

.06 

.007 

.067 

.504 

.45 

108.  When  fractions 

denoting 

tenths,  hundredths,  thou- 

sandths,  etc.,  are  expressed  like 

integers,  on 

the  decimal 

scale,  they 

are  said  to  I 

>e  expressed  decimally. 

Express 

decimally 

(22) 

(23) 

(24) 

(25) 

(26) 

t\ 

TT)\o 

IT  to" 

tV% 

Tciro 

i^ 

1000 

Ah 

TfoT 

A«A 

xh 

xioTT 

tV^ 

T^TT 

t¥A 

-ih 

T¥Tr 

t\%\ 

TTnrTT 

tMtt 

t\\ 

t'A 

tVA 

tWit 

A  , 
lOOO 

27.  What  is  the  name  of  the  third  decimal  order?    The 
fourth?    The  fifth?    The  sixth? 

28.  What  does  each  significant  figure  of  .0034  denote? 
Of  .00275  ?    Of  .03405  ?    Of  .000325  ?    Of  .030056  ? 


DECIMAL  FRACTIONS. 


75 


Copy  and  read 

(29) 

.246 

.0246 

.708 

.0708 

.3425 


(30) 

.0635 

.00635 

.3464 

.03464 

.32875 


(31) 

(32) 

00647 

.0307 

,000647 

.03007 

04056 

.030007 

004056 

.034005 

,32453 

.450605 

109.  When  a  decimal  fraction  is  expressed  decimally,  the 
right-hand  figure  is  written  in  the  order  indicated  by  the 
name  of  the  decimal.     Thus,  t^Wo¥  is  written  .00325. 


Express  decimally 

(33)  (34) 


Tiny 

8 

28 
TOTF-Q 

35(5 
TITTO' 


10000 

33 
TiyTFTTO" 

3  04  2 
TTTOlfO' 


(35) 

7 

1  ouiroir 


1  00000 


2_0  8 

lOOTTOlT 


3  0  5  6 
TTJTrUT)!}" 


380  4  5 
1  OO^OlT 


(36) 

2  9 

Tm^TJinriJ 

609  , 

TCi^oinnj" 

4  04  5 
T(n)T)(»0Tr 


3  3  033 
TT)¥U'OirO^ 

204056 
\-^ 


f-U^HWi 


Express  decimally 
(37) 
7  tenths ; 
24  hundredths ; 
29  thousandths ; 
405  thousandths; 
65  millionths; 
5064  millionths ; 
40056  millionths. 


(38) 
42  ten-thousandths; 
506  ten-thousandths ; 
4008  ten-thousandths ; 
65  hundred-thousandths ; 
6007  hundred-thousandths ; 
54008  hundred-thousandths 
3004  hundred-thousandths. 


39.  Eighty-five  thousandths. 

40.  Four  hundred  and  seven  thousandths. 

41.  Ninety-five  ten-thousandths. 

42.  Six  hundred  and  forty-four  ten-thousandths. 

43.  Seven  thousand  and  eighty-two  ten-thousandths. 

44.  Fifty-seven  hundred-thousandths. 

45.  Seven  hundred  and  eight  hundred-thousandths. 


7G  COMPLETE  ARITHMETIC. 

46.  Nine  thousand  and  forty-eight  hundred-thousandths. 

47.  Six  hundred  and  four  millionths. 

48.  Seven  thousand  six  hundred  and  forty-three  mill- 
ionths. 

49.  Forty  thousand  and  sixty-three  millionths. 

110.  An  integer  and  a  decimal  may  be  written  together 
as  one  number,  as  6y%  or  6.5;  25y^  or  25.07.  In  reading 
such  mixed  decimal  numbers,  the  integer  and  the  decimal 
are  connected  by  and.     Thus,  4.5  is  read  4  and  5  tenths. 

50.  Read45.-6;  30.25;  204.045;  84.0307. 

51.  Read  2005.045;  408.00075;  3040.0046;  50060.00705. 

52.  Read  400.045;  500.0063;  7000.0084;  60000.00006. 

Suggestion. — In  such  cases  read  the  integer  as  units;  as,  four 
hundred  units  and  forty-five  thousandths.  The  omission  of  the  word 
units  changes  the  mixed  number  to  a  pure  decimal. 

53.  Read  5600.0084;  40508.0307;  75000.000605. 

54.  Read  300000.000003;  35000000.000035. 

55.  Write  decimally  56ji^  ;   604yf |^ ;   400t^%Vo- 

56.  Write  decimally  207^^11^;  2560r^Vo%W 

57.  Three  hundred  units  and  three  hundred  and  forty- 
eight  millionths. 

DEFINITIONS,  PRINCIPLES,  AND  RULES. 

111.  A  Decimal  Fraction  is  a  fraction  whose  de- 
nominator is  some  power  of  ten. 

The  word  decimal  is  derived  from  decern,  a  Latin  word  meaning 
ten.  It  is  applied  to  this  class  of  fractions  because  the  successive 
fractional  units  or  orders  decrease  tenfold,  or  on  the  scale  of  ten. 

Note.— The  powers  of  ten  are  10,  100,  1000,  etc.     (Art.  388.) 

112.  Decimal  fractions  may  be  expressed  in  three  ways : 

1.  By  words;  as,  three  tenths,  twelve  hundredths. 

2.  By  writing  the  denominator  under  the  numerator,  in 
the  form  of  a  common  fraction ;  as,  f\,  j^J^^. 


DECIMAL  FRACTIONS.  77 

3.  By  omitting  the  denominator  and  writing  the  fraction 
in  a  decimal  form;  as,  .3,  .012.  The  denominator  is  un- 
derstood. 

Three  tenths,  j\,  and  .3,  each  express  the  same  decimal  fraction, 
but  the  term  decimal  is  usiuilly  applied  to  decimal  fractions  when 
expressed  by  the  third  method.  Since  common  fractions  may  have 
10,  100,  etc.,  for  a  denominator,  it  follows  that  decimal  fractions  are 
a  class  of  common  fractions. 

113.  The  Decimal  I^oint  is  a  period  placed  at  the 
left  of  the  order  of  tenths,  to  designate  the  decimal  orders. 

114.  A  Mixed  Decimal  is  a  decimal  ending  at  the 
right  with  a  common  fraction;  as,  .6f,  .033^. 

115.  A  Mixed  Decimal  Nuinher  is  an  integer  and 
a  decimal  written  together  as  one  number.  It  is  called 
more  simply  a  Mixed  Number. 

The  orders  on  the  left  of  the  decimal  point  are  integral, 
and  those  on  the  right  are  decimal.  The  decimal  orders  are 
called  Decimal  Places. 

116.  The  foHowing  table  gives  the  names  of  a  few  in- 
tegral and  decimal  orders,  and  shows  the  relation  between 
them: 


1  S'      Ml    .  -1       .:  2  ^  I      II 

000000000.00000000 

Integral  Orders.  Decimal  Orders. 

117.  Prlnciples. — 1.  The  denominator  of  a  decimal  frac- 
tion is  1  with  as  many  ciphers  annexed  as  there  are  decimol 
places  in  the  fraction.  -^ 


78  COMPLETE  ARITHMETIC. 

2.  The  value  of  the  successive  decimal  orders  decreases  tenfold 
from  left  to  right,  and  increases  tenfold  from  right  to  left. 
Hence, 

3.  The  removal  of  a  decimal  figure  one  place  to  the  right 
DECREASES  its  volue  tenfold,  and  its  removal  one  place  to  the 
left  INCREASES  its  valuc  tenfold. 

4.  The  name  of  a  decimal  is  the  same  as  the  name  of  its 
right-hand  order.     Hence, 

5.  A  decimal  is  read  precisely  as  it  woidd  he  were  the  denom- 
inator expressed. 

118.  Rules. — 1.  To  read  a  decimal.  Read  it  as  though  it 
were  an  integer,  and  add  the  name  of  the  right-hand  order. 

2.  To  write  a  decimal,  Write  it  as  an  integer,  and  so  place 
the  decimal  point  that  the  right-hand  figure  shall  stand  in  the 
order  denoted  by  the  name  of  the  decimal. 

Note. — When  the  number  does  not  fill  all  the  decimal  places, 
supply  the  deficiency  by  prefixing  decimal  ciphers. 

^WRITTEN  PROBLEMS. 

Express  decimally 

58.  Two  hundred  and  five  ten-thousandths. 

59.  Forty  thousand  and  thirty-four  millionths. 

60.  Two  thousand  and  four  hundred-thousandths. 

61.  Six  hundred  and  fifteen  ten-millionths. 

62.  Six  hundred  units  and  fifteen  ten-thousandths. 

63.  Fifteen  and  fifteen  thousandths. 

64.  Three  hundred  thousand  three  hundred  and  three 
hundred-millionths . 

65.  Five  million  and  eighty-five  ten-millionths. 

66.  Twelve  hundred-thousandths. 

67.  Four  hundred  units  and  four  hundred  and  sixty-fiv^e 
millionths. 

68.  Twenty-five  and  twenty-five  thousandths. 

69.  Five  thousand  units  and  five  thousandths. 

70.  Three  hundred  and  seventy-five  and  three  hundred 
and  seventy-five  billionths. 


REDUCTION  OF  DECIMALS.  79 

71.  Thirty  thousand  and  forty-six  hundred-thousandths. 

72.  One  million  and  forty-five  billionths. 

73.  Eighty  thousand  and  forty  and  three  hundred  and 
six  ten-thousandths. 

74.  Fifteen  thousand  units  and  fifteen  ten-thousandths. 

75.  Seventy-five  and  five  thousand  and  forty-three  mill- 
ionths. 

76.  One  million  units  and  one  millionth. 


REDUCTION  OF  DECIMALS. 
Case  I. 

IDeciixials  reclTicecl  to  IL<o^vex•  or  Higlier  Orders. 

1.  How  many  tenths  in  6  units?     In  15  units?     In  24 
units  ? 

2.  How  many  hundredths  in  5  tenths  ?    In  .6?   .8?   .7? 

3.  How  many  thousandths  in  .06?     In  .24?   .47?    .55? 

4.  How  many  tenths  in  .60?    In  .70?   .90?   .600?  .700? 
.800?  .5000?   1.50? 

5.  How  many  hundredths  in   .240?     In   .420?    .560? 
.4500?    .8500?   .35000?   .0700? 

-WBITTEN  PKOBLEMS. 

6.  Reduce  .875  to  millionths. 

Process  :    .875  =  .875000 

7.  Reduce  .0674  to  ten-millionths. 

^    8.  Reduce  .075  to  hundred-thousandths. 
9.  Reduce  62.7  to  thousandths. 

10.  Reduce  5.33  to  ten-thousandths. 

11.  Reduce  3.  to  hundredths. 

12.  Reduce  45.  to  ten-thousandths. 

13.  Reduce  .04500  to  thousandths. 

Process  :     .04500  =  .045 

14.  Reduce  5.24000  to  hundredths. 


80  COMPLETE    ARITHMETIC. 

119.  Principles. — 1.  Annexing  ciphers  to  a  decimal  frac- 
tion multiplies  both  of  its  terms  by  the  same  number,  and  hence- 
does  not  change  its  value.     (Art.  85.) 

2.  Catting  off  ciphers  from  the  right  of  a  decimal  fraction 
divides  both  of  its  terms  by  the  same  number,  and  Jwnce  does  not 
change  its  value.      (Art.  81.) 

Note. — The  annexing  of  decimal  ciphers  to  an  integer  does  not 
change  its  vahie.  Thus,  12.  ^-^  12.0,  or  12.00;  that  is,  12  units  =120 
tenths  =^  1200  hundredths,  etc. 


Case  II. 

DeciiTxals   reclnced   to   Coininon.  Fractions. 

15.  How  many  fifths  in  ^\?    j\?    .2?    .8? 

16.  How  many  fourths  in  yVo?  Toir?  t¥o?  -25?  .50? 
.75? 

17.  How  many  twentieths  in  yVV?  -y^?  .20?  .25?  .55? 
.75?    .95? 

^WRITTEN  PROBLEMS. 

18.  Reduce  .625  to  a  common  fraction  in  its  lowest  terms. 

Process  :     .625  =  j%%  =■- 1-|  =  f,  Ans. 
Reduce  to  common  fractions  in  lowest  terms 

19.  .125  25.  .004  31.  62.025 

20.  .75  26.  .5625  32.  37.625 

21.  .075  27.  .0125  33.  56.37| 

22.  .0625  28.  .3525  34.  247.33^ 

23.  .1625  29.  3.525  35.  16.66| 

24.  .2250  30.  37.75  36.  214.00^ 

120.  Rule. — To  reduce  a  decimal  to  a  common  fraction, 
Omit  the  decimal  point  and  supply  the  denominator,  and  then 
reduce  the  common  fraction  to  its  lowest  terms. 

Note. — When  the  denominator  is  written  the  fraction  is  both  deci- 
mal and  common. 


REDUCTION  OF  DECIMALS. 


81 


Case  III. 

CoiTimoia  IHractiorLS  redvicecl  to  DeciiTiuls. 


1  ? 


37.  How  many  tenths  in  |^? 

38.  How  many  hundredths  in  ^? 

39.  How  many  hundredths  in  -^^2  ^  ? 

40.  How  many  hundredths  in  2t^  A^ 


in  i^ 


3.? 
4  • 


2  ?        3  ? 
5  •         5  • 
3?        4  ^ 


7    ? 
20"- 

^5  • 


WRITTEN  PBOBLEMS. 

41.  Change  ^Ir  to  a  decimal. 


Process. 

125  )  3.00  (  .024,  Ans. 
2  50 
500 
500 


Since  j^-^  =  jl^  of  3,  and  since  3^3.000 
(Art.  119, "Note),  ji^  of  S  =  jl^  of  3.000  = 
.024.  Or,  j^^j^jl-^  of  3  units,  and  3  units 
=  3000  thousandths,  and  yf^  of  3000  thou- 
sandths =  24  thousandths  —  .024.  / 


Keduce  to  decimal  fractions 

48. 


42.  I 

43.  ^ 

44.  f\ 
45. 


47. 


"32 
64 

T23" 
80 

T23" 


49. 
50. 
51. 
52.    yV 


3  2 
2T 
87 
■2¥ 


04.       ^^ 


55. 
56. 
57. 
68. 
59. 


Too" 

2  3 
1250 


21 
T80 


60. 
61. 
62. 
63. 
64. 
65. 


12/^ 
25tIt 

1  4 
111 


121.  Rule. — To  reduce  a  common  fraction  to  a  decimal. 
Annex  decimal  ciphers  to  the  mimeratw  and  divide  hy  the 
denominator,  and  point  off  as  many  decimal  places  in  the  qxio- 
tient  as  there  are  annexed  ciphers. 


Notes. — 1.  When  a  sufficient  number  of  decimal  places  is  obtained, 
the  remainder  may  be  discarded,  or  the  quotient  may  be  expressed 
as  a  mixed  decimal. 

2.  When  the  denominator  of  a  common  fraction  in  its  lowest 
terms  contains  other  prime  factors  than  2  and  5,  the  process  will  not 
terminate. 

3.  When  the  quotient  repeats  the  same  figure,  or  the  same  set  of 
figures,  as  in  problems  63,  64,  and  65,  it  is  called  a  Repeating  Decimal, 
or  a  Circulating  Decimal,  and  the  figure  or  figures  repeated  are  called 
a  Repetend.  (Art.  431.) 


82  COMPLETE  ARITHMETIC. 

ADDITION  OF  DECIMALS. 

1.  Add  16.25,  48.037,  90.0033,  and  .864. 

Process.  Since  only  like  orders  can  be  added  (Art.  27), 

-.Q  25  write  the  figures  of  the  same  order  in  the  same 

48!o37  column.     Since  ten  units  of  any  order  make  one 

90.0033  unit  of  the  next  higher  order,  begin  at  the  right 

•^^^  and  add  as  in  simple  numbers.     Place  the  decimal 

155.1543,  Arts.  point  at  the  left  of  the  1  tenth. 

2.  Add  .375,  80.06,  45.0084,  .00755,  and  84.635. 

3.  Add  84.08,  16.075,  2.9,  1.96,  1.003,  and  5.0008. 

4.  Add  $15.34,  $65,048,  $9,083,  $12.,  $16.66|,  $18.06, 
$95,374,  and  $35.75. 

5.  Add  26.37^,  19.081    23.042i,  38.5,  6.00J,  and  7tV 

6.  Add  256  thousandths,  3005  millionths,  207  ten-thou- 
sandths, 34  ten-millionths,  and  94  hundred-thousandths. 

7.  Add  fifteen  thousandths,  eighty-one  ten-thousandths, 
fifty-six  millionths,  seventeen  ten-millionths,  and  two  hun- 
dred and  five  hundred-thousandths. 

8.  How  many  rods  of  fence  will  inclose  a  field,  the  four 
sides  of  which  are  respectively  46.6  rd.,  50.65  rd.,  24.33^ 
rd.,  and  27  rd.  ? 

9.  Five  bars  of  silver  weigh  respectively  .75  lb.,  1.15  lb., 
.86|^  lb.,  1.34  lb.,  and  .9  lb.:  what  is  their  total  weight? 

10.  The  average  amount  of  rain  in  San  Francisco  in  the 
winter  months  is  11.25  inches;  in  the  spring,  8.81  inches; 
in  the  summer,  .03  inches;  and  in  the  autumn,  2.75  inches : 
what  is  the  amount  for  the  year? 

122.  Rules. — To  add  decimals,  1.  Write  the  numbers  so 
that  figures  of  tJie  savie  order  shall  stand  in  the  same  column. 

2.  Add  as  in  the  addition  of  integers,  and  place  the  decimal 
point  at  tJie  left  of  Hie  tenths'  order  in  the  amount. 

Note. — If  a  mixed  decimal  does  not  contain  as  many  decimal 
places  as  either  of  the  other  numbers,  change  the  terminal  common 
fraction  to  a  decimal,  and  continue  the  division  until  the  requisite 
number  of  decimal  places  is  secured. 


1st  Process. 

2d  Process. 

47.625 
28.700 

47.625 

28.7 

18.925 

18!925 

2.  From  46.7  take  29.825. 

1st  Process. 

2d  Process. 

46.700 

29.825 

46.7 
29.825 

SUBTRACTlOxN  OF  DECIMALS.  83 

SUBTRACTION  OF  DECIMALS. 

1.  From  47.625  take  28.7. 

Reduce  the  decimals  to  a  like 
order  (Art.  119),  and  since  units 
can  only  be  taken  from  like  units, 
write  the  numbers  so  that  figures 
of  the  same  order  shall  stand  in 
the  same  column;  and  since  ten 
units  of  any  decimal  order  make 
one  unit  of  the  next  higher  order, 
subtract  as  in  simple  numbers. 
Place  the  decimal  point  at  the 
16.875  16.875  left  of  the  tenths'  order. 

Note. — A  comparison  of  the  two  processes  shows  that  it  is  unnec- 
essary to  fill  the  vacant  orders  with  ciphers. 

-  3.  From  4.05  take  2.0075. 

4.  From  .6 J  take  .00871 

5.  From  12.  take  .0005. 

6.  From  six  tenths  take  six  thousandths. 

7.  From  forty-four  thousandths  take  forty-four  millionths. 

8.  From  301  ten-thousandths  take  4005  millionths. 

9.  From  50065  ten-millionths  take  1307  billionths. 

10.  A  man  walked  33.7  miles  the  first  day  and  28.75 
miles  the  second:  how  much  farther  did  he  walk  the  first 
day  than  the  second  ? 

11.  The  average  amount  of  rain  at  Cincinnati  in  the 
summer  months  is  13.7  inches,  and  in  the  winter  months  it 
is  11.15  inches:   what  is  the  difference? 

12.  The  mean  height  of  the  barometer  at  Boston  is 
29.934  inches,  and  at  Pekin  it  is  30.154  inches:  what  is 
the  difference? 

123.  Rules. — To  subtract  decimals,  1.  Write  the  numbers 
so  that  figures  of  the  same  order  shall  stand  in  the  same  column. 

2.  Subtract  as  in  the  subtraction  of  integers,  and  place  the 
decimal  point  at  tlie  left  of  the  tenths'  order  in  the  remainder. 


84  COMPLETE  ARITHMETIC. 


MULTIPLICATION  OF  DECIMALS. 

1.  How  much  is  7  times  j^^^?    7  times  j\2    8  times  ^^? 

2.  How  much  is  8  times  j^-q  ?   8  times  j^  ?    6  times  yf^  ? 

3.  What  is  the  product  of  tV  X  tV?   toXt'o?  tif  X  t%? 

4.  What  is  the  product  of  j\  X  tU^    j\  X  rh^ 

5.  What  is  the  product  of  y|o  by  y^?    ^^  by  yf  ^  ? 

6.  What  is  the  denominator  of  the  product  when  tenths 
are  multiplied  by  units?  Tenths  by  tenths?  Tenths  by 
hundredths  ?  Hundredths  by  hundredths  ?  Hundredths  by 
thousandths  ? 

7.  What  is  the  denominator  of  the  product  of  any  two 
fractions  whose  denominators  are  powers  of  10? 

^^^ilITTEN   PKOBLEMS. 

8.  Multiply  .625  by  .43. 

Process.  gince  .625  =  tV^V,  and  .23  ■=  ^\\,  .625  X  .23  =  j%% 

*^23         ^  ^'^'^  ^  '^^^^  ^  '^^^J^\    ^"^^"''^'  -^^^  ^  '^^  ^  •^^^'^' 
— '-—  Since  thousandths  multiplied  by  hundredths  produce  hun- 

1250  dred-thoumndths,  the  product  contains  Jive  decimal  places, 

-.^or-c         or  as  many  as  both  of  the  factors. 

Multiply 

9.  6.5  by  .75        14.  4.36  by  .27        19.  .085  by  30. 

10.  .043  by  6.5         15.     64.  by  .032       20.  2.56  by  250. 

11.  .0432  by  5.4        16.  30.3  by  .018      21.  3.24  by  .334 

12.  .048  by  24.         17.   .056  by  24.        22.  5.75  by  8f  "^ 

13.  5.6  by  .056       18.     50.  by  .08         23.     16f  by  .045 

24.  Multiply  sixteen  thousand  by  sixteen  thousandths. 

25.  Multiply  205  millionths  by  46  thousandths. 

26.  Multiply  6.25  by  10.     By  100. 

Process.  Since  the  removal  of  a  decimal  figure  one 

fi2''V10    fi2 'S         place    to    the   left    increases   its   value   tenfold 

6  2^  V  100  —  62''  {^^T^t.  117,  Pr.  3),  the  removal  of  the  decimal 

point  one  place  to  the  right  multiplies  6.25  by 
10,  and  the  removal  of  the  point  two  places  to  the  right  multiplies 
6.25  by  100. 


DIVISION  OF  DECIMALS.  85 

27.  Multiply  3.406  by  100.     By  1000. 

28.  Multiply  .00048  by  1000.     By  100000. 

29.  Multiply  .0000256  by  10000.     By  1000000. 

PRINCIPLES  AND  RULES. 

124.  Principles. — 1.  Tlw  number  of  decimal  i^laees  in  the 
2)roduct  equals  the  numberr  of  decimal  'places  in  both  factors. 

2.  Each  removal  of  iJie  decimal  point  one  place  to  the  rigid, 
midtiplies  the  decimal  by  10. 

125.  Rules. — 1.  To  multiply  one  decimal  by  another, 
Multiply  as  in  tlie  multiplication  of  integers,  and  point  off  as 
many  decimal  places  in  tJie  product  as  there  are  decimal  places 
in  both  multiplicand  and  multiplier. 

Note. — If  there  be  not  enough  decimal  figures  in  the  prqduct, 
supply  the  deficiency  by  prefixing  decimal  ciphers. 

2.  To  multiply  a  decimal  by  10,  100,  1000,  etc..  Remove 
the  decimal  point  as  many  places  to  the  right  as  there  are  ciphers 
in  the  midtiplier. 

Note. — If  there  be  not  enough  decimal  places  in  the  product, 
supply  the  deficiency  by  annexing  ciphers, 

DIVISION  OF  DECIMALS. 

1.  How  many  times  are  5  tenths  contained  in  10  tenths? 
7  tenths  in  35  tenths? 

2.  How  many  times  are  7  hundredths  contained  in  21 
hundreths?     7  hundredths  in  35  hundredths? 

3.  What  is  A  ^  A?    T^V  -  tIt?    tU^^  tJ^tt? 

4.  What  is  .8  H-. 4?    .21-=- .07?    .084^.012? 

5.  What  is  tV  -  yh  ?    t'^It  -  rAtr  ?    tVV  -  tMt  ? 

Suggestion. — Keduce  the   fractions  to   a  common   denominator. 

6.  What  is  .3^.15?    .25---. 125?    .12-^.012? 

7.  Of  what  order  is  the  quotient  when  tenths  are  divided 
by  tenths  ?  Hundredths  by  hundredths  ?  Thousandths  by 
thousandths  ? 


SQ  COMPLETE   ARITHMETIC. 

8.  Of  what  order  is  the  quotient  when  any  order  is 
divided  by  a  like  order  ?  When  any  number  is  divided 
by  a  like  number? 

1?VRITTEW   PROBLEMS. 

9.  Divide  8.05  by  .35 

Process. 
.35  )  8.05  (  23.,  Ans.  35  hundredths  are  contained  in  805  Jiun- 

7  0  dredths,  a  like  number,  23  times,  and  hence 

1  05  8.05  -^  .35  =  23.     The  quotient  is  units. 

105 

10.  Divide  80.5  by  .35 

By   annexing    a    decimal    cipher  to   80.5, 
.35  )  80.50  (  230.,  Ans.     ^^i^-^^j^  ^^^^  ^^^^  change  its  value  (Art.  119), 

T7r7  the  dividend  and  divisor  are  made  like  num- 

jQ  5  bers,  and  hence  their  quotient  is  tinifs.     80.50 

-- ^  ^  .35  --=--  230. 

11.  Divide  .805  by  .35 

Process.  Since  .35  and  .80,  the  first  partial  dividend, 

.35  )  .805  (  2.3   Ans.  ^^®  l^ke  numbers,  the  first  quotient  figure  (2) 

70  denotes  units;  and  if  the  first  figure  denotes 

105  units,  the  second  must  denote  tenths.    Hence, 

105  .805  -^  .35  =-  2.3. 

The  pointing  in  all  the  cases  in  the  division  of  decimals,  may  also 
be  explained  on  the  principle,  that  the  dividend  is  the  product  of  the 
divisor  and  quotient,  and  hence  it  must  contain  as  many  decimal  places 
•  us  both  divisor  and  quotient. 

In  the  9th  example,  the  divisor  and  dividend  contain  an  equal 
number  of  decimal  places,  and  hence  there  are  no  decimal  places  in 
the  quotient. 

In  the  10th  example,  the  divisor  contains  one  more  decimal  place 
than  the  dividend,  and  hence  a  decimal  place  must  be  added  to  the 
dividend  before  the  division  is  possible. 

In  the  11th  example,  the  divisor  contains  two  decimal  places 
and  the  dividend  three,  and  hence  the  quotient  contains  one  decimal 
place. 


DIVISION  OF  FRACTIONS.  87 


Divide 

, 

12. 

32.4  by  1.8 

25. 

6.241  by  .0079 

13. 

2.56  by  .64 

26. 

67.5  by  .075 

14. 

.288  by  .036 

27. 

.675  by  75. 

15. 

82.5  by  2.75 

28. 

6.75  by  750. 

16. 

62.5  by  .025 

29. 

256.  by  .075 

17. 

9.  by  .45 

30. 

.256  by  250. 

18. 

4.53  by  .0302 

31. 

.0025  by  50. 

19. 

.3  by  .0125 

32. 

25.  by  .00125 

20. 

.625  by  12.5 

33. 

.001  by  100. 

21. 

.0256  by  .32 

34. 

100  by  .001 

22. 

17.595  by  8.5 

35. 

.045  by  900. 

23. 

3.3615  by  12.45 

36. 

$13.50  by  $.SH 

24. 

.031812  by  4.82 

37. 

$12.  by  $.06i 

38.  Divide  twenty-four  thousandths  by  sixteen  millionths. 

39.  Divide  seventy-eight  by  thirty-four  thousandths. 

40.  Divide  fifteen  millionths  by  six  hundredths. 

41.  Divide  45.7  by  10.    By  100. 

Process.  Since  the  removal  of  a  decimal  figure  one 

.  p.  -  ^^  1 A  _  4  r:7  place  to  the  right  decreases  its  value  tenfold 
.-\-^_H^^_  ^cy  (Art.  117,  Pr.  3),  the  removal  of  the  decimal 
point  one  place  to  the  left  divides  a  decimal  by 
10,  and  the  removal  of  the  point  two  places  to  the  left  divides  it 
by  100. 

42.  Divide  483.75  by  100.     By  1000. 

43.  Divide  54.50  by  100.     By  10000. 

44.  Divide  .005  by  1000.     By  100. 

PRINCIPLES  AND  KULES. 

126.  Principles. — 1.  Since  the  dividend  is  the  product 
of  the  divisor  and  quotient,  it  contains  as  many  decimal  places 
as  both  divisor  and  quotient.     Hence, 

2.  The  quotient  must  contain  as  many  decimal  places  as  tlw 
number  of  decimal  places  in  the  dividend  exceeds  the  number  of 
decimal  places  in  the  divisor.     Hence, 


88  COMPLETE  ARITHMETIC. 

3.  When  the  divisor  and  dividend  contain  the  same  number 
of  decimal  places,  the  quotient  is  units. 

4.  TJie  dividend  must  contain  as  many  decimal  places  as  the 
divisor  before  division  is  possible. 

5.  Each  removal  of  the  decimal  point  one  place  to  the  left 
divides  a  decimal  by  10. 

127.  Rules. — 1.  To  divide  one  decimal  by  another,  Divide 
as  in  the  division  of  integers,  and  point  off  as  many  decimal 
places  in  the  quotient  as  the  number  of  decimal  places  in  the 
dividend  exceeds  the  number  in  the  divisor. 

Notes. — 1.  When  the  divisor  contains  more  decimal  places  than 
the  dividend,  supply  the  deficiency  in  the  dividend  by  annexing  deci- 
mal ciphers. 

2.  When  the  quotient  has  not  enough  decimal  figures,  supply  the 
deficiency  by  prefixing  decimal  ciphers. 

3.  When  there  is  a  remainder,  the  division  may  be  continued  by 
annexing  ciphers,  each  cipher  thus  annexed  adding  one  decimal  place 
to  the  dividend.  Sufficient  accuracy  is  usually  secured  by  carrying 
the  division  to  four  or  five  decimal  places. 

2.  To  divide  a  decimal  by  10,  100,  1000,  etc..  Remove  the 
decimal  point  as  many  places  to  the  left  a^  Hiere  are  cipJiers  ii\ 
the  divisor. 

REVIEW   PKOBIiEMS. 

1.  Reduce  yf -  to  a  decimal. 

2.  Reduce  -^^^^  to  a  decimal. 

3.  Change  .325  to  a  common  fraction. 

4.  Change  .0045  to  a  common  fraction. 

5.  From  the  sum  of  67.5  and  .54  take  their  difference. 

6.  From  the  sum  of  64.5  and  .015  take  their  product. 

7.  Multiply  6.25  +  .075  by  6.25  — .075. 

8.  Divide  .0512  by  .032  X  .005. 

9.  From  25.6  -f-  .064  take  32.4  X  .015. 

10.  What  is  the  value  of  $5.33  X  2.5  --.075? 

11.  What  is  .08i  X  1.21  ^  .006^  X  .016? 

12.  Multiply  15  millionths  by  7  million. 

13.  Divide  16  ten-millionths  by  25  thousandths. 

14.  Divide  205  millions  by  41  ten-thousandths. 


UNITED  STATES  MONEY. 


89 


SECTION  X. 
UNITED  STATES  MONEY. 


PRELIMINARY  DEFINITIONS. 

128.  XInited  States 
Money  is  the  legal  cur- 
rency of  the  United  States. 
It  is  also  called  Federal 
Money. 

129.  The  denominations 
used  in  business  and  ac- 
counts, are  dollars^  cents,  and 
mills.  A  dollar  equals  100 
cents,  and  a  cent  equals  10 
mills. 

The  figures  denoting  dol- 
lars are  separated  from  those  denoting  cents  by  a  period, 
called  a  Separatrix  or  Decimal  Point,  and  they  are  preceded 
by  the  character,  $,  called  the  Dollar  Sign. 

130.  The  first  two  figures  at  the  right  of  dollars  denote 
cents,  and  the  third  figure  denotes  mills.  The  two  figures 
denoting  cents  express  hundredths  of  a  dollar,  and  the  figure 
denoting  mills  expresses  tenths  of  a  cent,  or  thousandths  of  a 
dollar.  The  three  figures  denoting  cents  and  mills  may  be 
read  together  as  so  many  thousandths  of  a  dollar. 

Notes. — 1.  United  States  Money  consists  of  Coin  and  Paper  Money. 
Coin  is  called  Specie  Currency  or  Specie,  and  paper  money  is  called 
Paper  Currency. 

2.  The  principal  gold  coins  are  the  fifty-dollar  piece,  double  eagle 
($20),  eagle  ($10),  half-eagle,  quarter-eagle,  three-dollar  piece,  and 
dollar. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dollar,  dime, 
half-dime,  and  three-cent  piece. 

The  nickel  coins  are  the  five-cent  piece,  three-cent  piece,  and  cent. 
C.Ar.-8 


9(J  COMPLETE  ARITHMETIC. 

The  copper  coins  (old)  are  the  two-cent  piece  and  cent. 

3.  Gold  and  silver  coins  are  alloyed,  to  make  them  harder  and 
more  durable.  The  gold  coins  contain  9  j^arts  of  gold  and  1  part  of 
an  alloy,  composed  of  copper  and  silver ;  and  the  silver  coins,  except 
the  three-cent  pieces,  contain  9  parts  of  silver  and  1  part  of  copper. 
Nickel  and  copper  coins  are  made  of  pure  metal. 

4.  Paper  money  consists  of  notes  issued  by  the  United  States, 
called  Treasury  Notes,  and  bank  notes  issued  by  banks. 

5.  Treasury  notes  of  a  value  less  than  $1,  as  fifty  cents,  twenty -five 
cents,  fifteen  cents,  ten  cents,  five  cents,  and  three  cents,  are  called 
Fractional  Currency. 


131.  NOTATION  AND  KEDUCTION. 

1.  Express  in  words,  $75.50;  $105.08;  $1000.45;  $15080.; 
$.o7;  $.o^o;  $o. 

2.  Express  in  words,  $37,507;  $250,075;  $80,005;  $.075; 
$2080.375;  $100,058;  $.065. 

3.  Read  decimally,  $70.25;  $140.05;  $387.60;  $560.09; 
$84.37;  $.08. 

4.  Read  decimally,  $.255;  $16,455;  $300,056;  $475,005; 
)5.375;  $240,061;  $.005. 

WRITTEN  PROBLEMS. 

5.  Write,  in  figures,  ten  dollars  fifty  cents, 

6.  Write  fiarty  dollars  sixty  cents  five  mills. 

7.  Write  100  dollars  37  cents  4  mills. 

8.  Write  430  dollars  5  cents ;  25  dollars  5  mills. 

9.  Write  75  cents  6  mills ;  6  cents  5  mills 

10.  Write  10  mills ;  10  cents  4  mills. 

11.  How  many  cents  in  $25?     $100?     $350? 

12.  How  many  mills  in  $47?     $150?     $165? 

13.  How  many  mills  in  $.75?     $.625?     $.017? 

14.  How  many  cents  in  $5.37?     $16.85?     $40.08? 

15.  How  many  mills  in  $.37^?     $4.62^?     $10? 

16.  Reduce  1500  cents  to  dollars. 

17.  Reduce  15000  mills  to  dollars. 

18.  Reduce  450  mills  to  cents. 

19.  Reduce  $25.08  to  mills. 

20.  Reduce  $100.01  to  cents;  to  mills. 


UNITED  STATES  MONEY.  91 


ADDITION  AND   SUBTRACTION. 

1.  A  man  paid  $7.50  for  a  pair  of  boots,  and  $5.50  for 
a  hat :  how  much  did  he  pay  for  both  ? 

2.  A  lady  paid  $15  for  a  shawl,  $5.75  for  a  hat,  $2.25 
for  a  pair  of  gloves,  and  $4  for  a  pair  of  gaiters :  what  was 
the  amount  of  her  purchases  ? 

3.  A  drover  bought  cows  at  $36.50  a  head,  and  sokl 
them  at  $40  a  head :  how  much  did  he  gain  ? 

4.  A  man  bought  a  coat  for  $24.25,  and  a  vest  for  $4.50, 
and  handed  the  merchant  three  $10  bills :  how  much  money 
did  he  receive  back? 

5.  A  mechanic  earns  $20  a  week,  and  his  family  expenses 
amount  to  $16.75  a  week:  how  much  has  he  left? 

6.  A  bookseller  bought  a  set  of  maps  for  $17,  and  a  set 
of  charts  for  $6.50,  and  sold  both  sets  for  $28.50:  how  much 
did  he  gain  ? 

WBITTEN  PROBLEMS. 

7.  What  is  the  sum  of  $.65,  $15.44,  $60. 62^,  $100, 
$94.05,  and  $.871? 

8.  From  $100.15  take  $62.37^^. 

9.  To  the  sum  of  $308.60  and  $190,125  add  their  dif- 
ference. 

10.  From  the  sum  of  $2750.  and  $1680.624^  take  their 
difference. 

11.  A  merchant's  sales  for  a  week  were  as  follows: 
Monday,  $125.60;  Tuesday,  $98.50;  Wednesday,  $190.30; 
Thursday,  $215.;  Friday,  $175.80;  Saturday,  $247.90:  what 
was  the  amount  of  his  sales  for  the  week? 

12.  A  man  exchanged  three  city  lots,  valued  respectively 
at  $900,  $1200,  and  $750,  for  a  farm  valued  at  $3075,  pay- 
ing the  difference  in  money :  how  much  money  did  he  pay  ? 

13.  A  man  receiving  a  salary  of  $1600  a  year,  pays  $325 
for  house  rent,  $450.80  for  provisions,  $200.60  for  clothing, 
and  $245  for  all  other  expenses :  how  much  has  he  left  ? 


92  COMPLETE  ARITHMETIC. 

14.  A  man  deposits  in  a  bank,  at  different  times,  $75, 
6230.80,  $180.40,  and  $95,  and  he  draws  out  $40,  $87.50, 
$331.45,  $20.15,  and  $18.60:  what  is  his  bank  balance? 

132.  Rule. — To  add  or  subtract  sums  of  money.  Write 
uniis  of  the  same  denomination  in  the  same  column^  add  or  sub- 
tract as  ill  simjde  numbers,  and  separate  dollars  and  cents  by  a 
period,  and  prefix  the  dollar  sign. 

MULTIPLICATION  AND  DIVISION. 

1.  A  mechanic  earns  $2.50  a  day:   how  much  will  he 
earn  in  6  days?     10  days? 

2.  What  will  8  barrels  of  flour  cost,  at  $7.25  a  barrel? 
At  $6.50  a  barrel? 

3.  What  will  20  yards  of  carpeting  cost,  at  $1.75  a  yard? 
At  $2.25  a  yard? 

4.  A  drover  paid  $38.70  for  9  sheep:  what  did  they  cost 
apiece  ? 

5.  A  man  paid  $42  for  8  tons  of  coal :  what  did  it  cost 
per  ton? 

6.  If  a  man  earn  $39  in  6  days :  how  much  will  he  earn 
in  10  days?     In  20  days? 

7.  At  25  cents  a  dozen,  how  many  dozens  of  eggs  can 
be  bought  for  $4.50? 

WBITTEN  PKOBLEMS. 

8.  A  farmer  sold  45  hogs  at  $22.45  apiece:  how  much 
did  he  receive  for  them  ? 

9.  A  miller  sold  237  pounds  of  flour,  at  $7.62^  a  barrel : 
what  amount  did  he  receive? 

10.  A  man  sold  a  farm  of  260  acres,  at  $33|-  per  acre : 
what  was  the  amount  received  ? 

11.  A  farm  containing  125  acres  was  sold  for  $5093.75: 
what  was  the  price  per  acre? 

12.  How  many  carriages,  at  $125  apiece,  can  be  bought 
for  $8000?     For  $7500? 


LEDGER  COLUMNS. 


93 


13.  At  $12,374-  a  ton,  how  many  tons  of  hay  can  be 
bought  for  $4653*^?     For  $1163.25  ? 

14.  A  farmer  sold  3  hogs,  weighing  respectively  278, 
309,  and  327  pounds,  at  $.07^-  a  pound :  how  much  did  he 
receive  ? 

15.  A  farmer  sold  in  one  year  536  pounds  of  butter,  at 
30  cts.  a  pound;  1200  pounds  of  cheese,  at  16f  cts. ;  and  19 
tons  of  hay,  at  $8.75  a  ton  :  how  much  did  he  receive? 

16.  A  grocer  bought  540  pounds  of  coffee  for  $81,  and 
420  pounds  of  tea  for  $525 ;  he  sold  the  coffee  at  18  cts.  a 
pound,  and  the  tea  at  $1.60  a  pound  :  how  much  did  he  gain  ? 

133.  Rules. — 1.  To  multiply  or  divide  sums  of  money  by 
an  abstract  number,  MulH'phj  or  divide  as  in  simple  numbers, 
separate  dollars  and  cents  in  the  result  by  a  period,  and  prefix 
the  dollar  sign. 

2.  To  divide  one  sum  of  money  by  another,  Reduce  both 
numbers  to  the  same  denomination,  and  divide  as  in  simple 
numbers. 

ABBREVIATED  METHODS. 


LEDGEK  COLUMNS. 

134.  A  Ledger  is  a  book 
in  which  business  men  keep 
a  summary  of  accounts. 

The  items  on  a  ledger  page 
often  make  long  columns  of 
figures,  which  are  added  or 
footed  with  absolute  accu- 
racy. 

135.  Let  the  pupil  foot  the 
following  ledger  columns  by 
adding  two  or  more  columns 
at  once,  being  as  careful  to 
obtain  accurate  results  as  he  would  be  in  actual  business. 
(See  Art.  22.) 


94  COMPLETE  ARITHMETIC. 


(1) 

$1.25 
8.14 

(2) 

$19.50 

20.00 

(3) 
$75.50 
184.30 

(4) 
$1912.88 
806.40 

2.75 

12.45 

111.10 

1000.00 

,65 

.75 

8.37 

14.52 
25.48 
40.50 

43.95 
263.55 
100.00 

1250.86 
943.82 
607.55 

12.50 

8.60 

90.00 

400.33 

4.65 

9.35 

7.15 

148.67 

.83 

.65 

13.48 

249.50 

7.16 

10.28 

1.20 

.73 

.84 

12.10 

2.75 

52.30 

900.25 

2040.00 
4508.70 
3406.30 

.95 

.48 
13.47 

.86 

.93 

2.95 

625.80 

314.87 

64.50 

1280.75 
•  1300.00 

877.77 

23.00 

14.63 

49.87 

620.14 

3.08 

9.82 

302.58 

8.60 

6.15 

12.60 

10.10 

7.45 

24.92 

19.30 

100.98 

13.33 

.83 

22.33 

78.60 

286.45 

.92 

.45 

14.8G 

9.81 

8.76 

12.57 

44.50 

77.88 

320.65 

1300.80 

1440.00 

986.70 

5.80 

18.19 

19.10 

87.80 

7.2G 
12.00 
5.00 
4.37 
6.45 

7.63 

14.60 

4.85 

9.63 

12.83 

8.50 
436,75 
135.20 

44.88 
65.90 

137.40 
1500.00 

885.73 

236.40 

13483.86 

17.83 

18.10 

6.01 

11456.20 

2.65 

7.63 

7.83 

88.00 

1.50 

2.20 

4.22 

24.30 

.85 

.35 

3,25 

16.50 

12.20 

.75 

.85 

9.85 

4,65 
8.15 

8.50 
4.65 

.62 
1.25 

100.00 
40.60 

Suggestion. — The  partial  footings  obtained  by  eacli  summary, 
should  be  written  upon  a  separate  piece  of  paper.  This  will  permit 
the  re-adding  of  any  column  or  set  of  columns,  as  the  case  may  be, 
without  the  trouble  of  re-adding  the  preceding  columns,  and  it  will 
also  avoid  tiie  defacing  of  the  page  by  erasures  and  corrections. 


ALIQUOTS.  95 


ALIQUOT  PARTS. 

136.  When  the  price  of  an  article  is  an  aliquot  part  of  a 
dollar,  the  cost  of  any  number  of  such  articles  may  be 
found  more  readily  than  by  multiplying. 

137.  The  aliquot  parts  of  a  dollar  commonly  used  in  busi- 
ness, are : 

50  cts.  =   I  of  $1.00  121  cts.  r=  I  of  $1.00 

25    ''    =:  i  of    1.00  61    "    --=  xV  of    1.00 

20    "    =  i  of    1.00  831-    *^    =  i  of    1.00 

10    ''    =  xV  of    1.00  16|    "    -=  i  of    1.00 

The  following  aliquot  parts  of  aliquot  parts  of  a  dollar 
are  frequently  used : 


25  cts.  =  \  of  50  cts. 

16|  cts.  =^  ^  of  33 1  cts. 

121  "  =  i  of  50  " 

12^  "  -  I  of  25   " 

6^  "  =\  of  50  " 

Q\    "  =\oi  25   " 

MENTAL   PROBLEMS. 

1.  What  will  56  pounds  of  grapes  cost,  at  12^  cts.  a  pound  ? 

Solution. — At  $1  a  pound,  56  pounds  will  cost  $56,  and  at  12  J 
cts.,  which  is  \  of  $1,  56  pounds  will  cost  \  of  $56,  which  is  $7. 

2.  What  will  120  spellers  cost,  at  25  cts.  apiece?     At 
33J  cts.  ? 

3.  What  is  the  cost  of  96  dozens  of  eggs,  at  16|  cts.  a 
dozen  ?     At  20  cts.  ?     At  25  cts.  ? 

4.  What  will  240  pounds  of  sugar  cost,  at  12^  cts.  a 
pound  ?     At  16f  cts.  ?     At  20  cts.  ? 

5.  At  16f  cents  a  dozen,  how  many  dozens  of  eggs  can 
be  bought  for  $15? 

Solution. — At  16f  cents  a  dozen,  $1  will  buy  6  dozens  of  eggs, 
and  $15  will  buy  15  times  6  dozens,  or  90  dozens. 

6.  At  12^  cts.  a  pound,  how  many  pounds  of  lard  can 
be  bought  for  S12?     For  $25? 

7.  How  many  pounds  of  butter,  at  33^  cts.  a  pound,  can 
be  bought  for  $15  ?     For  $33  ? 


96  COMPLETE  ARITHMETIC. 

8.  At  6J  cts.  a  quart,  how  many  quarts  of  currants  can 
be  bought  with  30  quarts  of  cherries,  at  10  cts.  a  quart? 

^VRITTEN   PKOBLEMS. 


9. 
a  yard 

What  will  348  yards  of  carpeting 

I? 

Process. 

cost, 

at  $1,624  cts. 

$1.62|  =  $l  +  50  cts. +  12 1 

cts. 

$348       =  cost  at  $1  a  yard. 
1       174       =    "     "    50  cts.  a  yard 
i        43.50  =    "     "    12J-  " 

$565.50==    "     ''    $1.62^ 

[. 

10. 
bushel 

What  will   1600  bushels  of  oats 
?     At  45  cts.  a  bushel?     At  62J- 

cost, 
cts.  ? 

at  374-  cts.  a 

11.  What  will  2464  bushels  of  wheat  cost,  at  $1.25  a 
bushel?     At  81.371?     At  $1,621? 

12.  What  will  1250  yards  of  carpeting  cost,  at  $1,374 
a  yard?    At  $1.50?     At  $1.87i? 

13.  What  will   640  bottles  of  ink  cost,  at  874  cents  a 
bottle  ?     At  621  cts.  ?     At  75  cts.  ? 

14.  At  25  cts.  a  dozen,  how  many  dozens  of  eggs  can  be 
bought  for  $42  ?    For  $105  ?     For  $60.50  ? 

15.  At  S3^  cts.  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $750?     For  $120? 

16.  What  will  5  lb.  10  oz.  of  butter  cost,  at  35  cts.  a 
pound  ? 

Process. 
$  .35    ==  cost  of  1  lb. 
$1.75  .  --=    "     "   5   " 

475  r=    «     «    8  oz.  (i  lb.) 

.044  =    "     "    2   "    (i  lb.) 
$1,969  =    "     "    5  lb,  10  oz. 

17.  What  will  9  lb.  13  oz.  of  cheese  cost,  at  15  cts.  a 
pound  ?     At  18  cts.  ?     At  20  cts.  ? 

18.  What  will   16  gal.  3  qt.  of  sirup  cost,  at  $1.75  a 
gallon ?     At  $1.62|?     At  $1.90 ? 


BILLS.  97 

19.  What  will  7  bu.  3  pk.  4  qt.  of  cherries  cost,  at  $4.25 
a  bushel?     At  $3.50?     At  $4.50? 

20.  What  will  2  pk.  7  qt.  of  chestnuts  cost,  at  $3.50  a 
bushel?    At  $2.75?    At  $2.62^?    At  $3,121? 

DEFINITION  AND  RULES. 

138.  An  Aliquot  Part  of  a  number  is  any  integer  or 
mixed  number  which  will  exactly  divide  it. 

139.  Rules. — 1.  To  find  the  cost  of  a  number  of  articles 
when  the  price  is  an  aliquot  part  of  a  dollar,  Find  the  cost 
at  $1,  and  take  the  aliquot  parts  of  the  residt. 

2.  To  find  the  number  of  articles  which  can  be  purchased 
for  a  given  sum  of  money  when  the  price  is  an  aliquot  part 
of  a  dollar,  Find  the  number  of  articles  that  can  be  purchased 
for  $1,  and  multiply  the  result  by  the  given  sum  of  money. 

BILLS. 

140.  Each  of  the  following  bills  should  be  neatly  made 
out  on  paper,  in  proper  form,  and  receipted. 


(1) 

Cincinnati,  0.,  Jan. 

1, 

1870. 

Thomas  Knight, 

1869                                           ^'^"^^^  'f 

Baker,  Smith 

&  Co. 

Nov.  18,  48  lb.  Castile  Soap,  @  16|c. 

. 

$8.00 

"      "     25    '"    Starch,             @     6^ 

. 

1.56 

"     30,  65    ''    Sugar,              @  15 

. 

. 

9.75 

"      "     33  gal.  Vinegar,          @,  20 

. 

6.60 

Dec.   12,  16  lb.  Eio  Coffee,      @  23 

. 

. 

3.68 

"      "      5    "    Star  Candles,  @  20 

. 

1.00 

"      ''     56    "    Butter,             @  33^ 

. 

. 

1.87 

"     15,  10   ''    Cheese,            @  15 

. 

1.50 

Received  Payment ^ 

$33.96 
Baker,  Smith  &  Co. 

per  CooNS. 

C.Ar.— 9 


98  COxMPLETE  ARITHMETIC. 

(2) 

St.  Louis,  March  3,  1870. 
James  Cooper  &  Bro., 

To  Charles  Camp  &  Co.,  Dr. 

To  37  bis.  Flour,  Ex.,       @  $4.50  .  .  .  .     $ 

"    23    "        "       Fj.,       @     5.25  ,      .  .  . 

25     "     Green  Apples,  @     2.12|  . 
14  bxs.  Lemons,             @     7.50 

5     "     Kaisins,  @,     4.75  ..... 


«     OK      a 


u        r      « 


Received  Payment, 
What  is  the  amount  due? 

(3) 

Cleveland,  O.,  Nov.  24,  1869. 
Dr.  William  Jones, 

To  Charles  C.  Wilhelm,  Dr 

To     24  Days'  Work,  @  $2.75       .  .  .  .     $ 

"      21  lb.  Nails,  @>         H  •  • 

"    540  ft.  Pine  Lumber,  @     4.50  per  100   .  .  . 

4  M.  Shingles,  @     8.33^ 


$ 


Cr. 

By  Cash,  Oct.  16,     .            .            .  '  .            .            .     ^S25 

"        "        "     23,          .            .            .  .            .             44 

'•'        "      Medical  Services  to  date  .            .            .15 


Received  Payment,  per  due-bill, 

Charles  C.  Wilhelm. 

What  is  the  amount  of  the  due-bill  ? 

4.  Henry  Smith  bought  of  John  Clarke,  of  Louisville, 
Ky.,  as  follows:  Mch.  10,  1870,  7  pair  calf  boots  @  $5.75; 
6  pair  ladies'  gaiters  @  $3.25;  10  pair  children's  shoes  @ 
$1.75;  Apr.  1st,  12  pair  coarse  boots  @  $3.12^;  6  pair  calf 


BILLS.  99 

shoes  @  S3. 30;  Apr.  12,  7  pair  ladies'  slippers,  $1.33|^;  3 
pair  calf  boots,  $d.G2^.  Make  out  and  receipt  the  above 
bill  as  clerk  of  John  Clarke. 

5.  Robert  Sterns  &  Co.  bought  of  Dudley  &  Bro.,  Detroit, 
Mich.,  Dec.  20,  1869,  as  follows:  5  doz.  ink-stands  @ 
$2. 12 J;  9  boxes  steel  pens  @  $.87^;  8  reams  note  paper  @ 
$3.50;  5  dozen  spellers  @  $2.33|^;  and  2  dozen  copy  books 
@  $1.80.  They  sold  Dudley  &  Bro.  3  sets  outline  maps  @. 
$8.25,  and  paid  them  $15  in  money.  Make  out  the  above 
bill  and  receipt  by  due-bill. 

6.  Mrs.  C.  B.  Jones  bought  of  Cole,  Steele  &  Co.,  of 
Indianapolis,  as  follows:  Nov.  12,  1869,  23  yds.  calico  @ 
16|c.;  45  yds.  sheeting  @  20c.;  Dec.  7th,  12  yds.  silk  @ 
$1,621;  8  handkerchiefs  @  45c.;  2  pair  kid  gloves  @  $1.87|-. 
Make  out  and  receipt  the  above  bill. 

DEFINITIONS. 

141.  An  Account  is  a  record  of  business  transactions 
between  two  parties,  with  specifications  of  debts  and  credits. 

The  party  owing  the  debts  specified,  is  called  the  Debtor, 
and  the  party  to  whom  they  are  due,  is  called  the  Creditor. 

142.  A  JSill  is  a  written  statement  of  an  account.  It  is 
drawn  by  the  creditor  against  the  debtor,  and  gives  the  time 
and  place  of  the  transaction,  and  the  names  of  the  parties. 

AVhen  the  debtor  has  made  payments  on  the  account,  or 
has  charges  against  the  creditor,  such  payments  or  charges 
are  called  Credits.     They  are  entered  as  in  Bill  3. 

143.  A  bill  is  receipted  by  writing  the  words  ^'Received 
Payment"  at  the  bottom,  and  afiixing  the  creditor's  name. 
This  may  be  done  by  the  creditor,  or  by  a  clerk,  agent,  or 
any  other  authorized  person. 

If  the  debtor  is  not  able  to  pay  a  bill  when  presented,  it 
may  be  accepted  by  writing  the  word  ''Accepted"  across  its 
face,  with  date  and  signature.     When  a  bill  is  paid  by  a 


100  COMPLEX K  ARITHMETIC. 

promissory  note  or  due-bill,  the  fact  may  be  added  to  the 
words  "Beceived  Payment"  as  in  Bill  3. 

144.  A  Bill  of  Goods  is  a  written  statement  of  goods 
sold,  with  the  amount  and  price  of  each  article,  and  the 
entire  cost.     It  is  also  called  an  Invoice. 

When  sales  are  made  at  different  times,  the  date  is 
written  at  the  left,  as  in  Bill  1. 


SECTION  XL 

MENSURATION, 

I.  SURFACES.— Definitions. 

145.  A  Line  is  length. 

146.  A  Straight  Line  is  a 


line  having  the  same  direction  throughout  its  whole  extent. 
Note. — The  word  line  is  commonly  used  to  denote  a  straight  line. 

147.  An  Angle  is  the  divergence  of  two  lines  meeting  at 
a  common  point.     The  point  of  meeting  is  called  the  vertex. 

A  Thus  the  divergence  of  the  lines  A  B 

and  A  C  is  the  angle  ABC,  and  the  point 
B  is  its  vertex. 

148.  When  a  line  so  meets  another  line  as  to  make  the 
two  adjacent  angles  equal,  each  angle  is  a  Right  Angle,  and 
the  first  line  is  perpendicular  to  the  second. 

A 

Thus  the  two  equal  adjacent  angles 
ABC  and  A  B  D  are  right  angles,  and 
the  line  A  B  is  perpendicular  to  the  line 
CD. 


MENSURATION. 


101 


149.  An  Obtuse  Angle  is  greater  than  a  right  angle, 
and  an  Acute  Angle  is  less  than  a  right  angle. 


Thus  the  angle  A  B  D  is  an  obtuse 
angle,  and  the  angle  ABC  is  an  acute 
angle.     The  line  A  B  is  an  oblique  line.  v, 


150.  A  Surface  is  that  which  has  length  and  width, 
but  not  depth  or  thickness. 

151.  A  Plane  Surface  is  a  surface  such  that  all 
possible  straight  lines  connecting  each  two  points  of  it,  lie 
wholly  within  the  surface.     It  is  also  called  a  Plane. 

Note. — To  determine  whether  the  surface  of  a  table  is  a  plane, 
take  a  ruler  with  a  straight  edge  and  apply  it  to  the  surface  in  many 
different  directions.  If  the  edge  rests  uniformly  upon  the  surface,  it 
is  a  plane. 


152.  A   Mectangle   is   a   plane 
figure  bounded  by  four  straight  lines      ^ 
and  having  four  right  angles. 


153.  A  Square  is  a  rectangle  with  its  four  sides 

A   Square  Inch   is    a   square  unch. 

each    side    of  which   is   an   inch    in 
length. 

The  figure  represents  a  square  inch  of 
real  size. 

A  square  foot,  square  yard,  square  rod,  etc., 
are  squares  whose  sides  are  respectively  1 
foot,  1  yard,  1  rod,  etc.,  in  length. 


154.  A  Triangle  is  a  plane  fig- 
ure bounded  by  three  straight  lines 
and  having  three  angles. 


equal. 


1  inch. 


102 


COMPLETE  ARITHMETIC. 


155.  A   Right-angled    Tri- 

(ingle  is  a  triangle  having  a  right 
angle.  One  of  the  sides  including 
the  right  angle  is  called  the  Base, 
and  the  other  the  Perpendicular  or 
Altitude. 


156.  A  Circle  is  a  portion  of  a  plane  bounded  by  a 
curved  line,  all  points  of  which  are 
equally  distant  from  a  point  within, 
called  the  center.  ' 

The   curved   line  which   bounds  a 
circle  is  its  Circumference. 

One-half  of  a  circumference  is  a  Semi- 
circumference;    one-fourth    is    a    Quadrant; 


and  any  portion  is  an  A 


re. 


157.  The  DiafUete?^  of  a  circle  is  a  straight  line 
passing  through  the  center  and  terminating  on  both  sides 
in  the  circumference.     One-half  of  a  diameter  is  a  Radius. 

All  the  diameters  of  a  circle  are  equal,  and  all  the  radii  are 
equal. 

The  circumference  of  a  circle  is  3.1416  (nearly  3i)  times  the 
diameter. 

158.  The  Avea  of  a  plane  figure  is  its  extent  of  surface, 
or  superficial  contents.  It  is  expressed  by  some  unit  of 
measure  as  a  square  inch,  a  square  foot,  etc. 

159.   The    area    of  a   right-angled 

triangle    is    one-half    the    area    of  a 

rectangle    with    the    same    base    and 

altitude.     The  triangle  A  B  C  is  one- 

c  half  of  the  rectangle  A  B  C  D. 

160.  The  area  of  a  circle  equals  the  product  of  the  cir- 
cumference by  the  one-half  of  the  radius. 


Note. — This  may  be  illustrated  by  dividing  a  circle  by  diameters 
into  eighths,  and  considering  each  a  triangle. 


MENSURATION.  '  103 

\ 

^  MENTAL    PKOBLEMS. 

1.  How  many  square  inches  in  a  piece  of  paper  4  inches 
.  long  and  1  inch  wide  ?     4  inches  long  and  2  inches  wide  ? 

2.  How  many  square  feet  in  a  piece  of  zinc  4  feet  long 
and  3  feet  wide  ?     4  feet  long  and  4  feet  wide  ? 

3.  How  many  square  inches  in  a  pane  of  glass  12  inches 
square  ?    Then  how  many  square  inches  in  a  square  foot  ? 

4.  How  many  square  feet  in  a  piece  of  oil-cloth  7  feet 
long  and  3  feet  wide?     8  ft.  long  and  6  ft.  wide? 

5.  How  many  square  feet  in  a  square  yard? 

6.  How"  many  square  feet  in  a  room  20  by  15  ft.  ?     30 
by  24  ft.  ?     50  by  30  ft.  ? 

Note.  —The  dhiiensions  of  a  plane  figure  are  usually  expressed  by 
writing  the  word  "  by,"  or  the  sign  "  X/'  between  the  figures  denoting 
the  length  and  width. 

7.  How  many  square  yards  in  a  pavement  40  by  5  yd.  ? 
50  X  4  yd.  ?     80  X  5  yd.  ? 

8.  How  many  square  miles  in  a  township  5  miles  square? 
6  miles  square? 

9.  How  many  square  inches  in  a  right-angled  triangle, 
whose  base  is  8  inches  and  whose  altitude  is  6  inches? 

10.  The  diameter  of  a  circle  is  10  feet:  what  is  its  cir- 
cumference ? 

^WKITTEN  PKOBLEMS. 

11.  How  many  square  feet  in  a  floor  374-  by  23  ft? 

12.  How  many  square  yards  in  a  walk  124.5  by  3.25  yd.? 

13.  How  many  square  feet  in  the  walls  of  a  room  24  by 
18f  ft.  and  10^  ft.  high?     What  is  the  area  of  the  ceiling? 

14.  How  many  square  chains  in  a  farm  134  chains  long 
and  52.5  chains  wide? 

15.  How  many  square  feet  in  a  city  lot  62\  ft.  front  by 
208  ft.  deep? 

16.  A  garden  containing  3267  square  yards  is  494-  yards 
wide :  hoAV  long  is  it  ? 


104  COMPLETE   ARITHMETIC. 

17.  A  street  containing  800  square  rods  is  33^  rods  long: 
how  wide  is  it? 

18.  How  many  yards  of  carpeting,  J  of  a  yard  wide,  will 
cover  a  room  15  by  8 J  yd.  ? 

19.  How  many  square  yards  in  a  triangular  garden  whose 
base  is  54.5  yards,  and  altitude  33.2  yards? 

20.  A   triangle   contains    270   sq.    in.,    and   the   base   is 
36  in. :   what  is  its  altitude  ? 

21.  The  diameter  of  a  circle  is   12   inches:    how  many 
square  inches  in  its  area  ? 

22.  How  many  square  feet  in  a  circle  whose  diameter  is 
20  feet  ? 

161.  Rules. — 1.  To  find  the  area  of  a  rectangle.  Multiply 
the  length  by  the  width. 

2.  To  find  either  side  of  a  rectangle.  Divide  the  area  by  tJie 
oilier  side. 

3.  To  find  the  area  of  a  triangle.  Multiply  the  base  by  one 
half  the  altitude. 

4.  To  find  the  area  of  a  circle.  Multiply  the  circumference 
by  one  fourth  of  the  diameter. 

Note. — The  two  dimensions   must  be  expressed  in  the  same  tie- 
nomination. 

II.  SOLIDS.  — Definitions. 

162.  A  Solid  is  that  which  has  length,  width,  and  depth 
or  thickness.     It  is  also  called  a  Volume  or  Body. 

A  line  has  only  length  ;    a  surface  has  length  and  width ;   and  a 
solid  has  length,  width,  and  depth. 


iiWfW'iii  163.  A  Bectanf/tilar  Solid  k 

a  body  bounded  by  six  rectangular 

surfaces. 

Illllllllllllllllllllllillliili'^i'^i 


The  surfaces  bounding  a  solid  are  called  Faces,  and  the  sides  of 
these  faces  are  called  Edges.  A  rectangular  solid  has  twelve  edges. 
The  face  on  which  a  solid  is  supposed  to  rest  is  called  its  Base. 


MENSURATION. 


105 


164.  A  Cube  is  a  body  bounded 
by  six  equal  squares.  All  its  edges 
are  equal. 

A  Cubic  Inch  is  a  cube  whose 
edges  are  each  one  inch  in  length. 


A  cubic  foot,  cubic  yard,  cubic   rod,  etc. 
are  each  cubes  whose  edges  are  respectively  1  foot,  1  yard,  1  rod,  etc. 

165.  A  Cylinder  is  a  solid  whose  two 
bases   are   equal   and   parallel   circles. 

166.  The  volume  of  a  body  is  called  its 
Solid  Contents,  or  Capacity.  It  is  expressed  in 
some  unit  of  measure,  as  a  cubic  inch,  a  cubic 
foot,  etc. 

MENTAL    PROBLEMS. 


1.  How  many  cubic  inches  in  a 
rectangular  solid,  4  inches  long,  1 
inch  wide,  and  1  inch  thick? 


2.  How  many  cubic  inches  in  a 
rectangular  solid,  4  inches  long,  3 
inches  wide,  and  1  inch  thick  ? 

3.  How  many  cubic  inches  in  a 
rectangular  solid,  4  inches  long,  3 
inches  wide,  and  2  inches  thick? 

4.  How  many  cubic  feet  in  a  block 
of  marble  6  ft.  long,  3  ft.  wide,  and 
2  ft.  thick?     10  ft.  long,  5  ft.  wide,  and  4  ft.  thick? 

5.  How  many  cubic  feet  in  a  cubic  yard? 

6.  How  many  cubic  feet  in  a  bin  6  ft.  long,  3  ft.  wide, 
and  3  ft.  deep?     8  ft.  long,  5  ft.  wide,  and  2  ft.  deep? 

7.  How  many  cubic  yards  in  a  room  5  yd.  long,  4  yd. 
wide,  and  3  yd.  high? 


106  COMPLETE  ARITHMETIC. 


WKITTEN  PROBLEMS. 

8.  How  many  cubic  feet  in   a  block  of  granite  16  ft. 
long,  8  ft.  wide,  and  5  ft.  thick? 

Process.  ^"^  block  IG  ft.  long,  1  it.  thick,  and  1  ft.  wide,  con- 
IP         £,  tains  16  cu.  ft.;  and  a  block  16  ft.  long,  1  ft.  thick, 

g      '     '  and  8  feet  wide,  contains  8  times  16  cu.  ft.,  or  128 

128  cu.  ft.  ^'^-  ^^'  i    ^"d  a  block  1 6  ft.  long,  8  feet  wide,  and  5  ft. 

5  thick,  contains  5  times  128  cu.  ft.  =  640  cu.  ft.    Hence, 

640  cu.  ft.  solid  contents  =  16  cu.  ft.  X  8  X  5. 

9.  How  many  cubic  feet  in  a  pile  of  wood  45  ft.  long, 
3^  ft.  wide,  and  7  ft.  high  ? 

10.  How  many  cubic  yards  in  a  cubic  rod? 

11.  How  many  cubic  feet  in  a  cube  each  of  whose  edges 
is  12i  ft.  in  length  ? 

12.  A  building,  65  ft.  by  44  ft.,  has  a  foundation  wall 
12  ft.  deep  and  2  ft.  thick:  how  many  cubic  feet  in  the 
foundations  ? 

13.  A  pile  of  wood,  containing  840  cu.  ft.,  is  30  ft.  long 
and  S^  ft.  wide :   how  high  is  the  pile  ? 

14.  If  27  bricks  make  a  cubic  foot,  how  many  bricks  will 
make  a  wall  45  ft.  long,  27  ft.  high,  and  2^  ft.  thick  ? 

15.  How  many  cans,  6  by  4  by  2  in.,  can  be  placed  in  a 
box  30  by  18  by  20  in  the  clear  ? 

16.  The  base  of  a  cylinder  is  12  inches  in  diameter,  and 
its  altitude  is  25  inches :  how  many  cubic  inches  in  its  solid 
contents  ? 

167.  Rules. — 1.  To  find  the  solid  contents  of  a  rectan- 
gular solid.  Multiply  the  length,  width,  and  thichiess  together. 

2.  To  find  the  length,  width,  or  thickness  of  a  rectangular 
solid,  Divide  Hie  solid  contents  by  the  product  of  the  other  two 
dimensions. 

Note. — The  three  dimensions  must  be  expressed  in  the  same  de- 
nomination. 

3.  To  find  the  solid  contents  of  a  cylinder.  Multiply  Hie 
area  of  tJie  base  by  the  altitude. 


REDUCTION. 


107 


SECTION   XII. 

DENOMINATE  NUMBERS. 


Wo&S^-^^-^^fi^ 


REDUCTION. 
Case  I. 

RecliAction  of  IDenoininate   Intesf^vs  and   IVLixecl 
TdiiTi"ber.s. 

1.  How   many  mills   in   9   cents?     In   12^^  cents?     62-^ 
cents?     100  cents? 

2.  How    many   cents    in    7    dimes?     25^   dimes?     45.4 
dimes?     56.8  dimes?     75.3  dimes? 

3.  How  many  dollars   in  50  dimes  ?     120  dimes  ?     145 
dimes?     1250  dimes?     1625  dimes? 

4.  How  many  dollars  in  800  cents  ?     2400  cents  ?     1365 
cents?     2235  cents? 

5.  How   many  farthings   in  9  pence  ?     72   pence  ?     90^ 
pence?     24.5  pence? 

Note. — For  tables  see  appendix. 

6.  How  many  pence  in   8|   shillings?     10^-  s.  ?     33 J  s.  ? 
2.5  s.  ?     6.5  s.  ? 


108  COMPLETE   ARITHMETIC. 

7.  How  many  shillings  in  15  £  ?     2.5  £?     16.4  £? 

8.  How  many  pence  in   22  far.  ?     48  far.  ?     105  far.  ? 
201  far.? 

9.  How  many    pounds    in    120  s.?     360  s.?     720  s.? 

10.  How    many    shillings    in.72d.  ?     144  d.?     25.2  d.? 
34.8  d.?     52.82  d.?     73.44  d.? 

11.  How  many  drams  in  8  oz.  avoir.?     20  oz.  ?    4.5  oz.? 

12.  How  many  ounces  in  5  lb.  avoir.  ?    10|  lb.  ?    2.5  lb.  ? 

13.  How  many  pounds  in  64  oz.  avoir.  ?   19.2  oz.  ?  4.8  oz.  ? 

14.  How  many  grains  in  5  pwt.  ?     10|^  pwt.  ?     2.5  pwt.  ? 

15.  How  many  pwt.  in  7  oz.  ?     6.5  oz.  ?     12-|  oz.  ? 

16.  How  many  ounces  of  gold  in  7  lb.  ?    12|  lb.  ?    1.5  lb.  ? 
4.5  1b.?     12.5  1b.? 

17.  How  many  pounds  of  gold   in   48   oz.  ?     14.4   oz.? 
2.52  oz.?     4.68  oz.?     62.4  oz.? 

18.  How  many  scruples  in  12  5  ?     8|  5  ?     14.5  5  ? 

19.  How  many  drams  in  15  B  ?     12J  S  ?     11.5  3  ? 

20.  How  many  ounces  in  9  lb  ?     5.5  lb  ?     10.5  lb  ? 

21.  How  many  inches  in  8 J  ft.  ?     15^  ft.  ?     33 J  ft.  ? 

22.  How  many  yards  in  12  rd.  ?     1.6  rd.  ?     3.2  rd.  ? 

23.  How  many  rods  in  11  yd.?     33  yd.?     6.6  yd.? 

24.  How  many  miles  in  18  fur.  ?     13.6  fur.  ?     7.2  fur.? 

25.  How  many  sq.  ft.  in  3^  sq.  yd.  ?     16|  sq.  yd.  ? 

26.  How  many  square  yards  in  12.6  sq.  ft.?    49.5  sq.  ft.? 
1.71  sq.ft.?     56.7  sq.ft.? 

27.  How  many  quarts  in  17  pk.  ?     12i  pk.  ?     301  pk.  ? 

28.  How  many  gallons  in  35  qt.  ?     14.8  qt.  ?     2.56  qt.  ? 

29.  How  many  weeks  in  365  days?     25.2  days? 

30.  How  many  years  in  192  mo.?     25.2  mo.?     100  mo.? 

WKITTEN   PROBLEMS. 

31.  Reduce  5£  6s.  3d.  to  pence;     1275  d.  to  pounds. 

o  £  6  s.  3d.  12  )  mo 

Process  :     _20  Process  :      20  )  106  3  d. 

■    106  8-  5£  Gs. 

12 

1275  d.,  Ans.  5  £  0  s.  3  d.,  An^. 


HEDUCnOX.  109 

32.  Reduce  38  lb.  11  oz.  7  dr.  to  drams. 

33.  Reduce  12  bu.  5  qt.  to  pints. 

34.  Reduce  13  mi.  5  fur.  3  yd.  to  yards. 

35.  Reduce  11  A.  3  R.  22  P.  to  perches. 

36.  Reduce  503  pt.  to  bushels. 

37.  Reduce  324  gi.  to  gallons. 

38.  Reduce  10280  ft.  to  miles. 

39.  Reduce  12460''  to  signs. 

40.  Reduce  30684  sec.  to  higher  denominations. 

41.  How  many  pence  in  £45?     In  £237|? 

42.  How  many  perches  in  95  A.  ?     320|  A.  ? 

43.  How  many  hundred-weight  in  4085  oz.  avoir.  ? 

44.  How  many  miles  in  12840  ft.? 

45.  Reduce  13  mi.  5^  fur.  to  inches.^ 

46.  Reduce  113420  inches  to  miles. 

47.  Reduce  3450  cubic  feet  of  wood  to  cords. 

48.  Reduce  5124  quarts  to  bushels. 

49.  Reduce  16  common  years  to  hours. 

50.  How  many  seconds  were  in  the  year  1868? 

51.  Reduce  4  common  yr.  45  d.  to  minutes. 

52.  Reduce  3.7  bushels  to  pints. 

53.  Reduce  4.5  rods  to  feet. 

54.  Reduce  3.65  lb.  Troy  to  ounces. 

55.  Reduce  15°  40'  36"  to  seconds. 

56.  Reduce  588487"  to  degrees. 

57.  Reduce  12.3  miles  to  feet. 

58.  Reduce  365^  days  to  weeks. 

59.  Reduce  706.35  perches  to  acres. 

60.  How  many  acres  in  12f  sq.  miles? 

168.  Rules. — I.  To  reduce  a  denominate  number  from 
a  higher  to  a  lower  denomination, 

1.  Multiply  the  number  of  the  highest  denomination  by  the 
number  of  units  of  the  next  lower  which  equah  a  unit  of  the 
higher,  and  to  the  product  add  the  number  of  the  lower  denomi- 
tiation,  if  any. 

2.  Proceed  in   like  m^anner  witli   this   and   each  successive 


110  COMPLETE  ARITHMETIC. 

result  tJms  obtained,  until  the  number  is  reduced  to  tJie  required 
denomination. 

Note. — The  successive  denominations  of  the  compound  number 
should  be  written  in  their  proper  orders,  and  the  vacant  denomina- 
tions, if  any,  filled  with  ciphers. 

II.  To  reduce  a  denominate  number  from  a  lower  to  a 
higher  denomination, 

1.  Divide  the  given  denominate  number  by  the  number  of 
units  of  its  oivn  denomination  which  equals  one  unit  of  the  next 
higher,  and  'place  the  remainder,  if  any,  at  the  right. 

2.  Proceed  in  like  manner  with  this  and  each  successive 
quotient  thus  obtained,  untiL  the  number  is  reduced  to  the  re- 
quired denomination. 

3.  The  last  quotient,  with  the  several  remainders  annexed  in 
proper  order,  will  be  the  answer  required. 

Note. — The  above  rules  also  apply  to  the  reduction  of  denominate 
fractions,  both  common  and  decimal.     (Art.  169.) 

Case   XL 

IRedvaction.  of  Denoiniiiate  yraotioiis. 

1.  What  part  of  a  peck  is  j^^  of  a  bushel  ?     ^  bu.  ? 

Solution. —  xV  hu.  =  xV  of  4  pk.  =  x\  pk.  or  ;}  pk.,  and  y\  bu.  = 
3  times  \  pk.  =  -J  pk.     Hence,  x\  bu.  —  |  pk. 

2.  What  part  of  a  quart  is  ^^  of  a  peck  ?     f^  pk.  ? 

3.  What  part  of  a  day  is  y\  of  a  week  ?     y^-  w.  ? 

4.  AVhat  part  of  an  hour  is  -^^  of  a  day  ?     -/^  d.  ? 

5.  What  part  of  an  inch  is  -^^^  of  a  foot  ?     y^  ft.  ? 

6.  What  decimal  part  of  an  inch  is  .03  of  a  foot? 

Solution.—  .03  ft.  =  .03  of  12  in,,  or  12  times  .03  in.  =  .36  in. 

7.  What  decimal  of  an  hour  is  .05  of  a  day?     .025  d.  ? 

8.  What  decimal  of  a  day  is  .12  of  a  week?    .012  w.  ? 

9.  What  decimal  of  a  quart  is  .125  of  a  peck?    .35  pk.  ? 

10.  What  part  of  an  inch  is  ^%  of  a  foot?    .08  ft.  ? 

11.  What  part  of  a  pint  is  -}q  of  a  gallon  ?    .06  gal.  ? 


REDUCTION.  Ill 

12.  What  part  of  a  foot  is  |  of  an  inch  ? 
Solution.—  f  in.  =  f  of  xV  ft.  =  ttV  ft. 

13.  What  part  of  a  week  is  ^j  of  a  day?     ^  d.  ? 

14.  What  part  of  an  hour  is  y^  ^^  ^  minute  ?     -\^  min.  ? 

15.  What  part  of  a  gallon  is  4  of  a  pint?     |  pt.  ? 

16.  What  part  of  a  pound  avoir,  is  f  of  an  ounce? 

17.  What  decimal  of  a  foot  is  .48  of  an  inch? 

Solution.—  .48  in.  =  .48  of  Jj  ft.  ^  j\  of  .48  ft.  =  .04  ft. 

18.  What  decimal  of  a  bushel  is  .12  of  a  peck?    3.6  pk.? 

19.  What  decimal  of  a  week  is  .49  of  a  day?    6.3  d.  ? 

20.  What  decimal  of  a  pound  Troy  is  .144  of  an  ounce? 
2.52  oz.?     38.4  oz.?    .72  oz.?     9.6  oz.? 

21.  What  decimal  of  a  ream  is  .8  of  a  quire?    2.8  quires? 

22.  What  part  of  a  dime  is  f  of  a  cent?    .625  ct. ? 

23.  What  part  of  a  shilling  is  |  of  a  penny?     .6  d.  ? 
.18  d.?     2.4  d.?     1.44  d.? 

24.  What  part  of  a  gallon  is  ||  of  a  pint?    .64  pt. ? 

^WKITTEN  PROBIiEMS. 

25.  Reduce  T^J^rir  ^^  ^  ^^J  to  the  fraction  of  a  minute. 

PROCESS :     -^  d.  =  IX  24  ^  _  TX^XAO  ^in.  ^  l^  „,i,, 
18000  18000  18000  25 

26.  Reduce  g^f^  of  a  pound  avoirdupois  to  the  fraction 
of  a  dram. 

27.  Reduce  Jf  of  a  yard  to  inches. 

28.  Reduce  j-J-  of  a  pound  Troy  to  pennyweights. 

29.  Reduce  .005  of  a  pound  to  the  decimal  of  a  penny. 

30.  Reduce  .0065  of  a  week  to  the  decimal  of  an  hour. 

31.  Reduce  9.6  pwt.  to  the  decimal  of  a  pound  Troy. 

32.  Reduce  3.96  inches  to  the  decimal  of  a  rod. 

33.  Reduce  30.8  rods  to  the  decimal  of  a  mile. 

34.  Reduce  .096  of  a  bushel  to  the  decimal  of  a  pint. 

35.  Reduce  |f  of  a  rod  to  the  fraction  of  a  league. 


112  COMPLETE    ARITHMETIC. 

36.  Reduce  ^  of  a  degree  to  the  fraction  of  a  circum- 
ference. 

37.  Reduce  ^l  of  a  day  to  the  fraction  of  a  minute. 

38.  Reduce  |-f^  of  a  week  to  the  decimal  of  an  hour. 

39.  Reduce  |  of  a  minute  to  the  fraction  of  a  day. 

40.  Reduce  11.2  perches  to  the  decimal  of  an  acre. 

41.  Reduce  13.62  cords  to  cord  feet. 

42.  Reduce  .037  lb.  avoirdupois  to  drams. 

43.  Reduce  56f  lb.  Troy  to  grains. 

44.  Reduce  y^^  of  a  gallon  to  the  fraction  of  a  pint. 

45.  Reduce  2.43  miles  to  feet. 

46.  Reduce  777.6  pence  to  pounds. 

47.  Reduce  1.408  ft.  to  the  decimal  of  a  mile. 

48.  Reduce  y^-  of  an  hour  to  the  fraction  of  a  day. 

49.  Reduce  .012  of  a  mile  to  yards. 

50.  Reduce  |  of  a  yard  to  the  decimal  of  a  mile. 

169.  Rule. — To  reduce  denominate  fractions  from  a 
higher  to  a  lower  denomination,  or  from  a  lower  to  a 
higher,   Proceed  as  in  tJie  reduction  of  denoioiinate  integers. 

Note. — Denominate  fractions  are  reduced  to  a  lower  denomination 
by  multiplying,  and  to  a  higher  denomination  by  dividing,  the  same 
as  denominate  integers ;  but  in  reduction  descending  there  are  no 
units  of  a  lower  order  to  add,  and  in  reduction  ascending  there  are 
no  remainders. 

Case  III. 

Reduction   of*   Denoininate   Fractions   to   Lowei* 
Inteyier:^. 

1.  How  many  months  in  |  of  a  year?  }  of  a  year? 
|-  of  a  year? 

2.  How  many  hours  in  |  of  a  day?  |  of  a  day?  \\ 
of  a  day? 

3.  How  many  minutes  in  ^  of  an  hour?  -^  of  an 
hour?     -^  of  an  hour? 

4.  How  many  yards  in  y^y  of  a  rod  ?  |^  of  a  rod  ?  -j^^ 
of  a  rod  ? 

5.  How  many  quarts  in  .75  of  a  peck?     1.25  pk.  ? 


REDUCTION.  113 

6.  How  many  months  in  .25  of  a  year?     .331  yr.  ? 

7.  How  many  days  in  .35  of  a  week?    4.5  w.?     7.3  w.? 

8.  How  many  pecks  and  quarts  in  .85  of  a  bushel? 

Solution.—  .85  bu.  =  .85  of  4  pk.  =  3.4  pk.,  and  .4  pk.  =::-  .4  of 
8  qt.  =  3.2  qt.     Hence,  .85  bu.  =  3  pk.  3.2  qt. 

9.  How  many  feet  and  inches  in  .75  of  a  yard? 

10.  How  many  quarts  and  pints  in  f  of  a  gallon? 

11.  How  many  days  and  hours  in  -J  of  a  Aveek? 

12.  How  many  pecks  and  quarts  in  .55  of  a  bushel? 

"WRITTEN   PROBLEMS. 

13.  Reduce  -f-^  of  a  day  and  .415  of  an  hour  each  to 
integers  of  lower  denominations. 

Process.  Process. 

7  V  24  -^^^  ^• 

/^  da.  =  t\  of  24  h.  =  ^-^^  h.  =:  lOi  h.  60 

^^  24.900  min. 

i  h.  =  I  of  60  rain.  =  ^  min.  =  30  min.  60 

2  54.000  sec. 

yV  da.  =  10  h.  30  min.  .415  h.  =  24  min.  54  sec. 

Reduce  to  integers  of  lower  denomination 

14.  -J  of  a  mile.  20.   .85  of  a  lb.  avoir. 

15.  3^  of  a  week.  21.   .325  of  a  ton. 

16.  T^^.  of  a  lb.  Troy.  22.   .081  of  a  yard. 

17.  H  of  a  rod.  23.   .9375  of  a  gallon. 

18.  -f^  of  an  acre.  24.   .5625  of  a  cwt. 

19.  i  of  a  cord.  25.  .0135  of  a  cord. 

170.  Rule. — To  reduce  a  denominate  fraction  to  inte- 
gers of  lower  denominations, 

1.  Multiply  the  fraction  by  the  number  of  units  of  the  next 
lower  denomination,  which  equals  a  unit  of  its  denomination. 

2.  Proceed  in  like  manner  with  the  fractional  part  of  the 
product  and  of  each  succeeding  product,  until  the  lowest 
denomination  is  reaxihed. 

C.Ar.— 10 


114  COMPLETE  ARITHMETIC. 

3.  The  integral  parts  of  the  several  products,  written  in 
proper  order,  will  he  the  lower  integers  sought. 

Note. — When  the  last  product  contains  a  fraction,  it  should  be 
united  with  the  integer  of  the  lowest  denomination,  forming  a  mixed 
number. 

Case  IV. 

PtecliactiorL   of   Integers  of  Lo^wer  Denonainations 
to   Fractions   of   Higlier  II)enoin.inations. 

1.  What  part  of  a  dollar  is  25  cents?     50  cts. ? 

2.  What  part  of  a  foot  is  8  inches  ?     10  in.  ? 

3.  What  part  of  a  day  is  9  hours?     15  h.? 

4.  What  part  of  a  yard  is  2  ft.  6  in.  ? 

Solution. —  1  yd.  =  36  in.,  and  2  ft.  6  in.  =  30  in. ;  1  in.  =^  j^-^  of 
a  yd.,  and  30  in.  =  |f  yd.  =  |  yd.     Hence,  2  ft.  6  in.  =  f  yd. 

5.  What  part  of  a  gallon  is  3  qt.  1  pt.  ? 

6.  What  part  of  a  bushel  is  2  pk.  5  qt.  ? 

7.  What  part  of  a  rod  is  3  yd.  2  ft.  ? 

8.  What  part  of  a  barrel  (31  gal.)  is  15  gal.  2  qt.  ? 

9.  What  part  of  3  pecks  is  2  pk.  4  qt.  ? 

10.  What  part  of  5  yards  is  2  yd.  2  ft.? 

Suggestion. — Each  of  the  above  answers  should  be  expressed 
both  as  a  common  fraction  and  as  a  decimal. 

WRITTEN  PROBLEMS. 

11.  Reduce  15  vv.  5  da.  to  the  fraction  of  a  common  year. 

Process. 
15  w.  5  da.  =r  110  da. 

iH  yr-  =  ft  yi-M  -i»i«. 

12.  Reduce  1  yd.  2  ft.  6  in.  to  the  fraction  of  a  rod. 

13.  Reduce  1  pk.  2  qt.  11  pt.  to  the  fraction  of  a  bushel. 

14.  Reduce  9  oz.  2^  dr.  to  the  fraction  of  a  pound. 

15.  Reduce  9  h.  36  min.  to  the  decimal  of  a  year. 

16.  Reduce  2  pk.  3*qt.  1.2  pt.  to  the  decimal  of  a  bushel. 

17.  Reduce  13  s.  4  d.  to  the  decimal  of  a  pound  Sterling. 


REDUCTION'.  115 

18.  Reduce  1  R.  14  P.  to  the  decimal  of  an  acre. 

19.  Reduce  8  oz.  8  pwt.  to  the  decimal  of  a  pound  Troy. 

20.  Reduce  1  fur.  18  rd.  1  yd.  to  the  decimal  of  a  mile. 

21.  What  part  of  1  bu.  3  pk.  is  5  pk.  6  qt.? 

22.  What  part  of  3  w.  4  da.  is  3  da.  8  h.  ? 

23.  What  part  of  12  A.  2  R.  is  1  A.  2  R.  10  P.  ? 

24.  What  part  of  3  barrels  of  flour  is  110  lb.  4  oz.  ? 

171.  Rule. — To  reduce  a  denominate  number,  simple 
or  compound,  to  the  fraction  of  a  higher  denomination, 
Reduce  the  number  which  is  a  part  and  the  number  which  is  a 
whole  to  the  same  denomination,  and  write  the  former  result  as 
a  numerator  and  the  latter  as  a  denominator  of  a  fraction. 

Notes. — 1.  The  arivswer  may  be  expressed  decimally  by  changing 
the  common  fraction  to  a  decimal. 

2.  When  the  whole  is  a  unit  and  the  part  a  compound  number, 
the  process  may  be  somewhat  shortened  by  reducing  the  number  of  the 
lowest  denomination  to  a  fraction  of  the  next  higher,  prefixing  the  higher 
number,  if  any,  and  then  reducing  this  result  to  a  fraction  of  the  next 
higher  denomination,  and  so  on,  until  the  required  fraction  u  reached. 
Thus,  in  the  16th  problem  above,  the  1.2  pt.  =  .6  qt, ;  and  3.G  qt.  =^ 
.45  pk.;  and  2.45  pk.=:.6125  bu. 

DEFINITIONS. 

172.  A  Denominate  Number  is  a  number  com- 
posed of  concrete  units  of  one  or  several  denominations. 
It  may  be  an  integer,  a  mixed  number,  or  a  fraction. 

173.  Denominate  numbers  are  either  Simple  or  Compound. 
A  Simple  Denominate  Number  is  composed  of 

units  of  the  same  denomination  ;  as,  7  quarts. 

A  Co^npound  Denominate  Nttniber  is  com- 
posed of  units  of  several  denominations ;  as,  5  bu.  3  pk.  7  qt. 
It  is  also  called  a  Compound  Number. 

Note. — Every  compound  number  is  necessarily  denominate. 

174.  Denominate  numbers  express  Currency,  Measure,  and 
Weight. 

Currency  is  the  circulating  medium  used  in  trade  and 
commerce  as  a  representative  of  value. 


116 


COMPLETE  ARITHMETIC. 


Measure  is  the  representation  of  extent,  capacity,  or 
amount. 

Weight  is  a  measure  of  the  force  called  gravity,  by 
which  bodies  are  drawn  toward  the  earth. 

175.  The  following  diagram  represents  the  three  general 
classes  of  denominate  numbers,  their  subdivisions,  and  the 
tables  included  under  each : 


T    n  f   1.  Cc 

I.  Currency,  \  ^   ^_ 


1.  Coin,  \     r  1.  United  States  Money, 

Paper  Money.  /     '-2.  English  Money. 


'  1.  Of  extension,  - 


II.  Measure, 


1.  Lines 
and  arcs, 


(  1.  Lo 
'  1  2.  Ci] 


1.  Long  Measure, 
Circular  Measure. 
2.  Surfaces :  Square  Measure. 


3.  Capacity, 


1.  Cubic  Measure, 

2.  Wood  Measure, 

3.  Dry  Measure, 

4.  Liquid  Measure. 


.  2.  Of  duration :  Time  Measure. 


III.  Weight 


•W 


1.  Avoirdupois  Weight, 

2.  Troy  Weight, 

3.  Apothecaries  Weight. 


Note. — For  tables  see  appendix. 

176.  The  Heduction  of  a  denominate  number  is  the 
process  of  changing  it  from  one  denomination  to  another 
without  altering  its  value. 

177.  Keduction  is  of  two  kinds,  Reduction  Descending  and 
Reduction  Ascending. 

JReduetion  Descending  is  the  process  of  changing 
a  denominate  number  from  a  higher  to  a  lower  denomi- 
nation. 

Heduction  Ascending  is  the  process  of  changing 
a  denominate  number  from  a  lower  to  a  higher  denomi- 
nation. 


REDUCTION.  117 


MENTAL  PROBLEMS. 


1.  How  many  half-pint  bottles  can   be  filled  with   2h 
gallons  of  sweet  oil  ? 

2.  A  boy  bought  f  of  a  bushel  of  chestnuts  for  $2,  and 
sold  them  at  10  cents  a  quart:  how  much  did  he  gain? 

3.  If  a  workman  can  do  a  job  of  work  in  120  hours, 
how  many  days  will  it  take  him  if  he  work  8  hours  a  day? 

4.  How  much  will  f  of  a  cwt.  of  sugar  cost,  at  16| 
cents  a  pound? 

5.  If  a  man  spend  J  of  each  day  in  sleep,  how  many 
hours  will  he  sleep  in  the  last  three  months  of  the  year? 

6.  If  a  man  walk  10  hours  a  day,  at  the  rate  of  3.3 
miles  an  hour,  how  far  will  he  walk  in  6  days? 

7.  How  many  square  inches  in  the  surface  of  a  brick  8 
inches  long,  4  inches  wide,  and  2  inches  thick? 

8.  How  many  square  feet  in  a  board  12.6  ft.  long  and 
8  inches  wide? 

9.  How  many  solid  feet  in  a  plank  16  feet  long,  1^  feet 
wide,  and  4  inches  thick? 

10.  A  man  paid  $36  for  a  stack  of  hay  containing  4^ 
tons,  and  sold  it  at  50  cents  a  hundred :  how  much  did  he 


gain? 


'WRITTEN  PROBLEMS. 


11.  How  many  yards  of  carpeting,  f  of  a  yard  wide,  will 
carpet  a  room  27  feet  long  and  21^  feet  wide? 

12.  How  many  acres  in  a  street  2^  miles  long  and  5  rods 
wide  ? 

13.  What  would  be  the  cost  of  a  township  of  land  6 
miles  square,  at  $10.50  an  acre? 

14.  A  rectangular  field  is  60  rods  long  and  37|^  rods 
wide:  how  many  boards,  each  12  feet  long,  will  inclose  it 
with  a  fence  5  boards  high? 

15.  At  $5.62|-  a  cord,  what  will  be  the  cost  of  a  pile  of 
wood  85  ft.  6  in.  long,  6  ft.  4  in.  high,  and  4  ft.  wide? 


118  COMPLETE  ARITHMETIC. 

16.  How  many  bricks,  4  by  8  in.,  will  it  take  to  pave  a 
walk  16  feet  wide  and  6 J  rods  long? 

17.  How  many  gold  rings,  each  weighing  3.2  pwt.,  can 
be  made  from  a  bar  of  gold  weighing  .75  of  a  pound? 

18.  An  octavo  book  contains  480  pages:  how  many  reams 
of  paper  will  it  take  to  print  an  edition  of  1200  copies, 
making  no  allowance  for  waste  ? 

19.  How  many  perches  of  masonry  in  the  wall  of  a  cellar 
45  feet  long,  34  feet  wide,  and  2-J-  feet  thick  ? 

Note. — In  measuring  walls  of  cellars  and  buildings,  masons  take 
the  distance  round  the  outside  of  the  walls  (the  girth)  for  the  length, 
thus  measuring  each  corner  twice. 

20.  How  many  perches  of  stone  in  the  walls  of  a  fort 
120  feet  square,  the  walls  being  33J-  feet  high  and,  on  an 
average,  11  feet  thick? 

21.  What  will  it  cost  to  excavate  a  cellar  40  ft.  long, 
21  ft.  6  in.  wide,  and  4  ft.  deep,  at  $1.75  a  cubic  yard? 

22.  A  bin  is  8  ft.  long,  3^  ft.  wide,  and  4  ft.  deep :  how- 
many  bushels  of  grain  (2150|  cu.  in.)  will  it  hold? 

23.  A  circular  park  is  165  yards  in  diameter:  how  many 
acres  does  it  contain  ? 

24.  How  many  cubic  feet  in  the  capacity  of  a  circular 
well  3^  ft.  in  diameter  and  20  ft.  deep? 

25.  A  cylindrical  cistern  is  5  ft.  in  diameter  and  6  ft. 
4  in.  deep :  how  many  gallons  of  water  will  it  hold  ? 

26.  A  congressional  township  is  6  miles  square,  and  is 
divided  into  36  sections :  how  many  acres  in  a  section  ? 

27.  A  tract  of  land  is  4  miles  long  and  2^  miles  wide : 
how  many  sections  does  it  contain  ?     How  many  acres  ? 

28.  A  speculator  bought  S\  sections  of  land  at  $4.50  an 
acre,  and  sold  them  at  $6.25  an  acre:  how  much  did  he 
gain? 

29.  A  man  sold  a  farm  containing  a  quarter  of  a  section 
of  land,  for  $3280 :  what  did  he  receive  per  acre  ? 


THE  METRIC  SYSTEM. 


119 


THE   METRIC   SYSTEM. 

178.  The  Metric  Syste7n  is  a  system  of  weights  and 
measures  based  on  the  decimal  scale. 

The  system  was  first  adopted  by  France,  and  it  is  now  in  general 
use  in  nearly  all  the  countries  of  Europe.  The  use  of  the  system 
in  the  United  States  was  legalized  by  Congress  in  1866,  and  it  is 
employed,  to  some  extent,  in  several  departments  of  the  government 
service.     It  has  long  been  used  by  the  Coast  Survey. 

The  convenience  and  accuracy  of  the  system  have  secured  its  very 
general  adoption  in  the  sciences  and  arts,  but  it  is  not  probable  that 
it  will  soon  come  into  general  use  in  business  transactions. 

179.  The  Meter  is  the  primary  unit  of  the  system.  It 
is  the  ten-millionth  part  of  the  distance  on  the  earth's  surface 
from    the   equator  to   tlie 

pole. 

The  Liter  (le'-ter)  is 
the  unit  of  the  measures 
of  capacity.  It  is  the 
thousandth  part  of  a  cubic 
meter. 

The  Gram  is  the  unit 
of  weights.  It  is  the  weight 
of  the  thousandth  part  of 
a  liter  of  water  at  its  great- 
est density. 

180.  The  meter,  liter,  and  gram  are  each  multiplied  hy 
10,  100,  1000,  and  10000,  giving  multiple  units,  and  they 
are  also  each  divided  by  10,  100,  1000,  giving  the  decimal 
subdivisions  of  tenths,  hundredths,  thousandths,  etc. 

181.  The  multiples  are  named  by  prefixing  to  the  name  of 
the  primary  unit,  or  base,  the  Greek  numerals,  I)eka  (10), 
Hedo  (100),  KUo  (1000),  and  Myria  (10000);  and  the  subdi- 
visions are  named  by  prefixing  the  Latin  words,  Deci  (10th), 
a}Ui  (100th),  and  Milli  (1000th). 


120 


COMPLETE  ARITHMETIC. 


METRIC  TABLES. 
182. — I.  Measures  of  Length. 


i_ 


i 


The  Unit  is  a 

M 

ETER  —  39.37  inches,  nearly. 

Denominations. 

Values.                Equivalents. 

Myriameter 

3rz 

lOOOO  meters  =  6.2137  mi. 

Kilometer 

= 

1000  meters  ^  0.6214  mi. 

Hectometer 

■■^^ 

100  meters  ==  328iV  ft. 

Decameter 

— 

10  meters  =  393.7  in. 

Meter 

= 

1  meter    =  39.37  in. 

Decimeter 

— 

.1  meter   ^  3.937  in. 

Centimeter 

= 

.01  meter    =  0.3937  in. 

Millimeter 

== 

.001  meter    =  0.0394  in. 

Decimal  Scale. 

1  i 
i  i 

a 

o 

Decameter. 
Meter. 

Decimeter. 
Centimeter. 
Millimeter. 

0    0 

0 

0    0.000 

Ten  units  of  any  denomination  of  the  above 
table  equal  one  unit  of  the  next  higher  de- 
nomination, and,  hence,  the  successive  de- 
nominations correspond  to  successive  orders 
of  figures  in  the  decimal  system :  the  meter 
denoting  units ;  the  decameter,  tens,  etc. 

The  correspondence  between  the  metric 
denominations  and  those  of  United  States 
Money  is  also  noticeable.  The  millimete)' 
corresponds  to  mills;  the  centimeter  to  cents; 
the  decimeter  to  dimes;  the  metei'  to  dollars,  etc. 

The  above  diagram  shows  that  a  decimeter  is  a  little  less  than  four 
inches,  and  that  a  centimeter  is  a  little  more  than  f  of  an  inch. 

Note. — As  no  abbreviations  for  the  names  of  the  metric  units 
have  been  agreed  upon  in  this  country,  the  names  are  given  in  full 
in  this  work.  The  tables  of  equivalents  need  not  be  memorized  by 
the  pnpil. 


THE  METRIC  SYSTEM.  121 

183. — II.  Measures  of  Surface. 

The  Unit  is  an  Are,  or  a  Square  Decameter. 

Denominations.  Values.  Equivalents. 

Hectare    =  10000  sq.  meters  =  2.471  acres. 
Are  (air)  -—      100  sq.  meters  =  119.6  sq.  yards. 
Centiare    =  1  sq.  meter    --  1.196  sq.  yards. 

Since  100  units  of  each  denomination  in  the  above  Decimal  Scale. 

table  equal  one  of  the  next  higher,  each  occujDies  two  o                   £ 

orders  of  figures.     The  centiares  correspond,  in  this  |       ^          "^ 

respect,  to  cents,  which  occupy  two  places.  K      <          o 

,  ,      .  -,    .  .11  0    0    0.00 

The  above  table  is  used  in  measuring  land. 

The  primary  unit  for  the  measuring  of  small  surfaces  is  a  square 
meter. 

Note. — Centiare  is  also  written  Centare. 

184. — III.  Measures  of  Capacity. 

The  Unit  is  a  Liter,  or  a  Cubic  Decimeter. 

„  ,  Equivalents. 

Denominations.  \alues.  rv      ^r  r-     -7  tit 

Dry  Measure.  Liquid  3[eastire. 

Kiloliter  =  1000  liters  =  1.308  cu.  yd.    =  264.17  gallons. 
Hectoliter  =    100  liters  =  2.8375  bu.         =  26.417  gallons. 

Decaliter  =  10  liters  rz=  9.08  qt.              =  2.6417  gallons. 

Liter  =  1  liter    =  0.908  qt.            -=:  1.0567  quarts. 

Deciliter  =  .1  liter    =  6.1022  cu.  in.  =  0.845  gill. 

Centiliter  =  .01  liter    =  0.6102  cu.  in.  =  0.338  fluid  ounce. 

Milliliter  =  .001  liter  .  r--^  0.061  cu.  in.     =0.27  fluid  dram. 

The  kiloliter  equals  a  cubic  meter,  the  liter  a  cubic  decimeter,  and 
the  milliter  a  cubic  centimeter. 

The  Kiloliter  is  called  a  Stere  (Stair),  and  is  the  principal  measure 
of  wood,  stone,  etc.  One-tenth  of  a  stere  is  a  Decistere,  and  10  steres 
are  a  Decastere. 

The  liter  is  used  in  measuring  liquids,  and  the  hectoliter  in  meas- 
uring grains. 

Note. — When  the  tliree  dimensions  of  a  regular  solid  are  expressed 
in  decimeters,  their  product  will  be  the  contents  in  liters. 
r.Ar.-ll. 


122 


COMPLETE  ARITHMETIC. 


185.— IV.  Weights. 


The 

Unit 

is  a 

Gram  =  15.432  grains. 

Denominations. 

Values.            iMiuivalonts  in  Av.  Wci^ri,t. 

Millier,  or  tonneau 

I  ^^ 

1000000  grams  =.  2204.G  iwiinds 

Quintal 

^=^ 

100000  grams  =  220.4G  pounds 

Myriagram 

-_-= 

10000  grams  =.  22.046  pounds 

Kilogram,  or 

Kilo 

— 

1000  grams  ^  2.2046  pounds 

Hectogram 

r= 

100  grams  =  0.5274  ounces 

Decagram 

= 

10  grams  —  0.3527  ounce 

Gram 

= 

1  gram    —  15.432  gr.  Troy. 

Decigram 

= 

.1  gram    —  1.5432  gr.  Troy. 

Centigram 

= 

.01  gram    ^  0.1543  gr.  Troy. 

Milligram 

= 

.001  gram    ^  0.0154  gr.  Trov. 

A  millier  equals  tlic  weight  of  a  cubic  meter  of  water  at  its  greatest 
density;  a  kilogram  equals  a  liter  of  water;  the  gram,  a  cubic  centi- 
meter of  water;,  and  the  milligram,  a  cubic  millimeter  of  water.  The 
kilogram,  called,  for  brevity.  Kilo,  is  the  ordinary  weight  of  commerce. 


186. — V.  Metric  Equivalents  of  Common 
Denominations. 


Long  Measure. 

An  inch  =  .0254  meter. 
A  foot     =  .3048  meter. 
A  yard    =  .9144  meter. 
A  rod      =  5.029  meters. 
A  mile    -=  1.6094  kilometers. 

Square  Measure. 

A  sq.  inch  =  .000645  sq.  meter. 
A  sq.  foot    =  .0929  sq.  meter. 
A  sq.  yard  =  .8362  sq.  meter. 
A  sq.  rod    =r  .2529  are. 
An  acre       ■=  .4047  hectare. 
A  sq.  mile  =  259  hectares. 


Cubic  Measure. 

A  cu.  inch  ^=r  .0164  liter. 
A  cu.  foot   —  .2832  hectoliter. 
A  cu,  yard  •=  .7646  stere. 
A  cord         =  3.625  i^tcrcs. 

Weight. 

A  grain  =^  .0648  gram. 

A  pound  av.     =  .4536  kilogram. 
A  pound  Troy  ■=  .373  kilogram. 
A  ton  =  .907  tonncau. 


A  gallon  =^  3.786  litems. 

A  bushel  ^^  .3524  hectoliters. 


The  new  nickel  5  cent  piece  weighs  5  grams,  and  is  2  centimeters, 
or  A  of  a  meter,  in  diameter. 


THE  METRIC  SYSTEM.  123 


MENTAL   PROBLEMS. 

1.  How  many  meters  in  a  decameter?    In  a  hectometer? 
A  kilometer?     A  myriaraeter? 

2.  What  part  of  a  meter  is  a  decimeter  ?    A  centimeter  ? 
A  millimeter? 

3.  Name  the  metric  units  of  length,  in  order,  from  the 
lowest  to  the  highest, 

4.  How  many  liters  in  a  hectoliter  ?    In  a  decaliter  ?    A 
kiloliter  ? 

5.  Name  the  metric  units  of  capacity  from  the  highest 
to  the  lowest? 

6.  What  part  oF  a  gram  is  a  centigram  ?    A  decigram  ? 
A  milligram  ? 

7.  How  many  meters  in  5  decameters?    44  decameters? 
225  decameters  ?     34.6  decameters? 

8.  How  many  liters  in  3   hectoliters  ?     37  hectoliters  ? 
22.5  hectoliters?     7.45  hectoliters? 

9.  How  many  grams   in  8  kilograms  ?     24  kilograms  ? 
3.25  kilograms?    .456  kilogram? 

10.  What  decimal  part  of  a  meter  is  a  centimeter?  15 
centimeters  ?     72  centimeters  ? 

11.  What  decimal  part  of  a  gram  is  a  milligram?  24 
milligrarns?     245  milligrams? 

12.  When  the  metric  units  are  expressed  on  the  decimal 
scale,  which  order,  from  the  decimal  point,  is  the  decimeter? 
The  millimeter?     The  centimeter? 

13.  Which  order,  from  the  decimal  point,  is  the  deca- 
liter?    The  kiloliter?     The  hectoliter? 

14.  Read  the  several  orders  in  324.56  meters  as  metric 
units. 

Alls.  3  hectometers  2  decameters  4  meters  5  decimeters  and  6 
centimeters. 

15.  Read  the  several  orders  in  504.046  grams  as  metric 
units. 

16.  Read  4080.57  liters  in  metric  units. 


124  COMPLETE  ARITHMETIC. 

-WKITTEN   PROBLEMS. 

17.  Write  5  kilograms  7  hectograms  6  decagrams  5  grams 
and  6  centigrams  on  the  decimal  scale  as  grams. 

Ans-  5765.06  grams. 

18.  Write  6  hectoliters  4  decaliters  3  liters  and  5  deci- 
liters on  the  decimal  scale  as  liters. 

19.  How  many  meters  in  6  kilometers  7  decameters  and 
5  decimeters? 

20.  How  many  grams  in  6  kilograms  4  decagrams  and  8 
centigrams  ? 

21.  Reduce  234.56  hectograms  to  grams. 

Process  :     234.56  X  100  =  23456.     Ans.  23456  grams. 

22.  Reduce  345.8  centigrams  to  grams. 
Process  :     345.8  -^  100  =  3.458.     Ans.  3.458  grams. 

23.  Reduce  45.06  Idloliters  to  liters. 

24.  Reduce  35.4  hectoliters  to  liters. 

25.  Reduce  84.5  ares  to  square  meters. 

26.  Reduce  132.4  centimeters  to  meters. 

27.  Reduce  24000  millimeters  to  meters. 

28.  Reduce  434.5  centiliters  to  liters. 

29.  Reduce  3.225  quintals  to  grams. 

30.  Reduce  746.35  decagrams  to  kilograms. 

31.  How  many  yards  in  220  meters? 

Process  :    39.37  in.  X  220  --  12  -h-  3  ==  240.59  +.    Ans.  240.59  yd. 

32.  How  many  miles  in  44.5  kilometers? 

33.  How  many  inches  in  24  centimeters  ? 

34.  How  many  bushels  in  250  hectoliters  of  wheat? 

35.  How  many  gallons  in  37|  liters  of  sirup  ? 

36.  How  many  pounds  of  butter  in  150  kilos? 

37.  How  many  liters  in  35  cubic  feet? 

38.  How  many  steres  in  20  cords  of  wood? 

39.  How  many  ares  in  f  of  ian  acre? 

40.  How  many  meters  in  1760  feet? 


cwt. 

lb. 

oz. 

dr. 

7 

44 

6 

11.5 

12 

13 

0 

7.6 

23 

56 

12 

0. 

27 

00 

14 

8.4 

COMPOUND  NUMBERS.  125 

SECTION  XIII. 
COMPOUND  NUMBERS. 

ADDITION  AND  SUBTRACTION. 

1.  What  is  the  sum  of  7  cwt.  44  lb.  6  oz.  11.5  dr.;  12  cwt. 
13  lb.  7.6  dr.;  23  cwt.  56  lb.  12  oz. ;  and  27  cAvt.  14  oz. 
8.4  dr.? 

Since  only  like  numbers  can  be  added, 
Process.  write  the  numbers  of  the  same  denomina- 

tion in  the  same  columns.  The  sum  of  the 
drams  is  27.5  dr.  =  1  oz.  11.5  dr.  Write 
the  11.5  dr.  under  drams,  and  add  tlie  1  oz. 
with  the  ounces.     Proceed  in  like  manner 

3  T.  10      15       1     11.5       "ntil  the  numbers  of  the  several  denomi- 

nations are  added. 

2.  Add  16  -mi.  7  fur.  27  rd.  3  yd.  2  ft.  8i  in. ;    18  mi. 

4  fur.  5  yd.   1  ft.  7f  in. ;    27  mi.  35  rd.  4  yd.  5^  in. ;   and 
6  fur.  24  rd.  3  yd.  2  ft. 

3.  Add  13  w.  6d.  13 h.  48min.;  8w.  13 h.  51  min.  37  sec; 
12 w.  5d.  22 h.  16  min.  44  sec;  1  w.  10 h.  15  min.;  and  Id. 
10  h.  26  sec. 

4.  Add  24  lb.  10  oz.  17  pwt.  22  gr.;  16  lb.  19  pwt.;  10  oz. 
15  pwt.  21  gr.;  45  lb.  9  oz.  18  gr.;  and  13  lb.  11  oz.  18  pwt. 
23  gr. 

5.  Add  15  bu.  3  pk.  7  qt. ;  27  bu.  5  qt.  1  pt. ;  8  bu. 
2  pk.  1  pt. ;  47  bu.  3  pk. ;  12  bu.  2  pk.  1  qt.  1  pt. ;  and 
19  bu.  1  pk.  3  qt. 

6.  Add  16°  32'  43";  28°  47'  53";  25°  53";  4  s.  48'  48"; 
lis.  16°  36'  59";  and  5s.  18°  7'  8". 

7.  How  many  cords  of  wood  in  three  piles,  the  first  being 
23  ft.  long,  4  ft.  wide,  and  7  ft.  high ;  the  second,  28  ft. 
long,  4  ft.  wide,  and  6|  ft.  high;  and  the  third,  17  ft.  long, 
8  ft.  wide,  and  7^  ft.  high  ? 


Pkocess, 

Id. 

13 

9 

yd. 
3 
4 

ft. 
1 

0 

in. 

6.4 

11.5 

3 

^ 

0 

=  1 

6.9 

6 

126  COMPLETE  ARITHMETIC. 

8.  From  13  rd.  3  yd.  1  ft.  6.4  in.  take  9  rd.  4  yd.  11.5  in. 

Write  the  subtrahend  under  the  minuend, 
pLicing  the  numbers  of  the  several  denomina- 
tions in  columns,  as  in  compound  addition. 
Since  11.5  in.  is  greater  than  6.4  in.,  add  12  in. 
to  6.4  in.  and  tlien  subtract.  To  balance  the 
12  in.  added  to  the  minuend,  add  1  ft.  (12  in.) 
3      4         2       .9  to  the  subtrahend  (Art.  30,  Pr.  3),  or,  if  pre- 

ferred, subtract  1  ft.  from  the  minuend. 
Proceed  in  like  manner  until  the  difference  between  the  numbers 
of  the  several  denominations  is  found.     Reduce  the  |  yd.  to  feet  and 
inches,  and  add  the  result  to  the  0  ft.  6.9  in.  of  the  remainder. 

9.  From  30  mi.  6  fur.  14  rd.  3  yd.  1  ft.  4  in.  take  25  mi. 
36  rd.  4  yd.  2  ft.  10  in. 

10.  From  33  rd.  1  yd.  2  ft.  11  in.  take  16  rd.  3  yd.  8  in. 

11.  From  104°  11'  20'^  take  83°  43'  36". 

12.  Boston  is  71°  4'  9"  W.  longitude,  and  San  Francisco 
is  122°  26'  15"  AV.  longitude :  what  is  their  difference  in 
longitude  ? 

13.  From  the  sum  of  245  A.  2  R.  27  P.  and  187  A.  3  R. 
34  P.  take  their  difference. 

14.  A  note  was  given  July  23,  1863,  and  it  was  paid  Nov. 
16,  1868 :  how  long  did  it  run  ? 

15.  A  man  was  born  Sept.  12,  1827,  and  his  eldest  son 
was  born  Apr.  6,  1855:  what  is  the  difference  in  their  ages? 

16.  Baltimore  is  situated  76°  37'  W.,  and  Vienna  16° 
23'  E. :  what  is  their  difference  in  longitude  ? 

17.  A  ship  in  latitude  37°  20'  north,  sails  15°  45'  south  ; 
then  12°  36'  north;  then  18°  40'  south:  what  is  her  latitude ? 

DEFINITIONS  AND  RULES. 

187.  A  Compound  dumber  is  a  number  composed 
of  units  of  several  denominations.     (Art.  173.) 

188.  Compound  numbers  are  of  the  same  kind  when  their 
corresponding  terms  denote  units  of  the  same  denomination ; 
as,  3  bu.  2  pk.,  and  6  bu.  3  pk.  5  qt. 


COMPOUND  NUMBERS.  127 

189.  Compound  Addition  is  the  process  of  finding 
the  sum  of  two  or  more  compound  numbers  of  the  same 
kind. 

190.  Rule. — To  add  compound  numbers, 

1.  Write  the  compound  numbers  to  he  added  so  tJiat  units  of 
the  same  denomination  shall  stand  in  tlie  same  column. 

2.  Add  the  colum^n  of  tlie  lowest  denomination^  and  divide 
the  sum  by  the  number  of  units  of  that  denomination,  ivhich 
equals  a  unit  of  the  next  higher  denomination ;  write  the  re- 
mainder under  the  column  added,  and  add  the  quotient  wiHi 
the  next  column. 

3.  Li  like  manner  add  the  remaining  columns,  writing  the 
sum  of  the  highest  column  under  it. 

Note. — In  both  simple  and  compound  addition,  the  sum  of  each 
column  is  divided  by  the  number  of  units  of  that  denomination,  ivhick 
equals  one  of  the  next  higher  denomination.  In  simple  addition  this 
divisor  is  10 ;  in  compound  addition  it  is  a  varying  number,  since 
the  several  denominations  are  expressed  on  a  varying  scale. 

191.  Compound  Subtraction  is  the  process  of 
finding  the  diflTerence  between  two  compound  numbers  of 
the  same  kind. 

192.  KuLE. — To  subtract  one  compound  number  from 
another, 

1.  Write  the  subtrahend  under  the  minuend,  jAacing  terms 
of  the  same  denomination  in  the  same  column. 

2.  Beginning  at  the  right,  subtract  each  successive  term  of  the 
subtrahend  from  the  corresponding  term  of  the  minuend,  and 
write  the  difference  beneath. 

3.  If  any  term  of  the  subtrahend  be  greater  than  the  corre- 
sponding term  of  the  minuend,  add  to  the  term  of  the  minuend 
as  many  units  of  that  denomination  as  equal  one  of  the  next 
higher,  and  from  the  sum  subtract  the  term  of  the  subtrahend, 
writing  the  difference  beneath. 

4.  Add  one  to  the  next  term  of  the  subtrahend,  and  proceed 
as  before. 

Note.— Instead  of  adding  one  to  the  next  term  of  the  subtrahend, 
one  may  be  subtracted  from  the  next  term  of  the  minuend. 


128  COMPLETE  ARITHMETIC. 


MULTIPLICATION  AND  DIVISION. 

1.  Multiply  15  rd.  3  yd.  1  ft.  7  in.  by  11. 

Process.  Since   the  value   of  llie  units   of 

1'   'c\    S   -1    1  ft   7  in        the  successive  denominations  increases 

*  11  from  right  to  left,  begin  at  the  right 

4  fur.  11  rd.  5  yd.  2  ft.  5  in.       h^n^.     11  times  7  in.  r=-  77  in.  =  6  ft. 

5  in.  Write  the  5  in.  under  inches, 
and  reserve  the  6  ft.  to  add  witli  the  product  of  feet.  Proceed  in 
like  manner  until  the  numbers  of  the  several  denominations  are 
multiplied. 

2.  If  a  man  can  build  7  rd.  11  ft.  6  in.  of  fence  in  a 
day,  how  much  can  15  men  build? 

3.  How  many  bushels  of  wheat  in  18  bins,  each  con- 
taining 124  bu.  3  pk.  5  qt.  ? 

4.  How  much  hay  in  13  stacks,  each   containing  4  T. 
13  cwt.  56  lb.  ? 

5.  What  is  the  weight  of  12  silver  spoons,  each  weigh- 
ing 2  oz.  13  pwt.  14  gr.  ? 

6.  Divide  19  mi.  4  fur.  20  rd.  2  yd.  9  in.  by  7. 

Process.  Since  the  value  of  the 

7  )  19  mi,  4  fur.  20  rd.  2  yd.  0  ft.  9  in.       "nits  of  the  successive  de- 

2  mi.  6  fur.  14  rd.  1  yd.  2  ft.  81  in.      nominations  decreases  from 

left  to  right,  begin  at  the 
left  hand.  }  of  19  mi.  =  2  mi.  with  5  mi.  remaining.  Write  the 
2  mi.  under  miles,  and  reduce  the  5  mi.  to  furlongs,  and  add  tlie 
4  fur.  which  gives  44  fur.  }  of  44  fur.  =  6  fur.  with  2  fur.  remain- 
ing. Keduce  the  2  fur.  to  rods,  add  the  20  rd,,  take  }  of  the  result, 
and  proceed  in  like  manner  until  the  numbers  of  all  the  denomina- 
tions are  divided. 

7.  Divide  27  mi.  3  fur.  25  rd.  12  ft.  6  in.  by  12. 

8.  A  ship  sailed  39°  12'  40"  in  21  days:  how  many  de- 
grees did  it  average  each  day? 

9.  If  15  equal  bars  of  silver  contain  24  lb.  8  oz.  16  pwt., 
what  is  the  weight  of  each  bar  ? 

10.  If  12  equal  bins  hold  430  bu.  2  pk.  of  wheat,  how 
much  wheat  is  there  in  each  bin  ? 


COMPOUND  NUiMBERS.  129 

11.  From  13  w.  5d.  18  h.  40  min.  take  7  w.  23  h.  45  min., 
and  divide  the  difference  by  15. 

12.  Add  4  fur.  23  rd.  3  yd.  2  ft.  and  7  fur.  16  rd.  1  ft., 
and  divide  the  sum  by  22. 

13.  From  the  sum  of  56  lb.  13  oz.  9  dr.  and  47  lb.  15  oz. 
15  dr.  take  their  difference,  and  divide  the  result  by  9. 

14.  How  many  rings,  each  weighing  4  pwt.  15  gr.,  can  be 
made  from  a  bar  of  gold  weighing  1  lb.  10  oz.  ? 

Suggestion.— Eeduce  both  divisor  and  dividend  to  the  same  de- 
nomination. 

15.  How  many  kegs,  each  containing  5  gal.  1  qt.,  can  be 
filled  from  a  cask  holding  63  gal.  ? 

16.  How  many  rotations  will  a  wheel  12  ft.  6  in.  in  cir- 
cumference make  in  rolling  |  of  a  mile? 

17.  How  many  lengths  of  fence,  each  11  ft.  6  in.,  will 
inclose  a  square  field  each  side  of  which  is  20  rd.  5  yd.  ? 

18.  How  many  barrels,  each  holding  2  bu.  3  pk.,  will 
hold  132  bushels  of  apples? 

19.  How  many  axes,  each  weighing  3  lb.  3  oz.,  can  be 
made  from  a  ton  of  iron? 

20.  How  many  steps,  2  ft.  6  in.  each,  will  a  man  take 
in  walking  round  a  field  45  rods  square? 

21.  The  length  of  a  solar  year  is  365  d.  5  h.  48  min. 
48  sec. :  how  much  time  is  ^^  of  a  solar  year  ? 

DEFINITIONS  AND  RULES. 

193.  Compound  Multiplication  is  the  process  of 
taking  a  compound  number  a  given  number  of  times.  The 
multiplier  is  always  an  abstract  number. 

194.  Rule. — To  multiply  a  compound  number, 

1.  Write  the  multiplier  under  the  loivest  denomination  of  the 
midtiplicand. 

2.  Beginning  at  tJie  right,  midtiply  each  term  of  the  midti- 
plicand  in  order,  and  reduce  each  product  to  the  next  higher 


130  COMPLETE  ARITHMETIC. 

denomination,  writing  the  remainder  under  tJie  term  multiplied, 
and  adding  the  quotient  to  the  next  jyroduct. 

Note. — In  both  simple  and  compound  multiplication,  the  successive 
products  are  each  divided  by  the  number  of  units  of  their  denomination, 
which  equals  one  of  the  next  hie/her  denomination. 

195.  Conijwiind  Division  is  the  process  of  dividing 
a  compound  number  into  equal  parts. 

196.  Rules. — I.  To  divide  a  compound  number, 

1.  Write  the  divisor  at  the  left  of  the  dividend,  as  in  simple 
division. 

2.  Beginning  at  the  left,  divide  each  term  of  the  dividend  in 
order,  and  write  the  quotient  under  the  term  divided. 

3.  If  the  division  of  any  term  give  a  remainder,  reduce  tJie 
remainder  to  the  next  lower  denonmiation,  to  the  result  add  the 
number  of  thai  denomination  in  the  divide7id,  and  then  divide 
as  above. 

Note. — When  the  divisor  is  a  large  number,  the  successive  terms 
of  the  quotient  may  be  written  at  the  right  of  the  dividend,  as  in 
long  division. 

II.  To  divide  a  compound  number  by  another  of  the 
same  kind.  Reduce  both  compound  numbers  to  tJie  same  de- 
nomination, and  then  divide  as  in  simple  division. 

Note. — This  is  not  properly  compound  division,  since  the  com- 
pound numbers  are  reduced  to  simple  numbers  before  dividing. 


LONGITUDE  AND  TIME. 

197.  Lofigitiide  is  distance  east  or  west  from  a  given 
meridian.  It  is  measured  in  degrees,  minutes,  and  seconds. 
Thus,  15°  24'  40"  east  longitude  denotes  a  position  15°  24' 
40"  east  of  the  meridian  from  which  longitude  is  reckoned. 

vSince  every  circle  is  divided  into  360  degrees,  the  length  of  a 
degree  depends  upon  the  size  of  the  circle  of  which  it  is  a  part. 

The  length  of  a  degree  of  longitude  depends  upon  the  latitude  of 
the  parallel  on  which  it  is  measured.  It  is  greatest  at  the  equator, 
where  it  is  09^  miles;  and  least  at  the  poles,  where  it  is  nothing. 


LONGITUDE  AND  TIME. 


131 


198.  The  earth  rotates  on  its  axis  from  west  to  east 
once  every  twenty- four 
hours,  and  the  illumi- 
nated space  between  any 
two  meridians  passes  un- 
der the  sun's  rays  in  the 
same  length  of  time.  A 
degree  of  surface  at  the 
equator  passes  under  the 
sun's  rays  in  the  same 
time  as  a  degree  at  any 
latitude  between  the  equator  and  the  polar  circle. 

199^  When  the  vertical  rays  of  the  sun  are  on  the 
meridian  of  any  place,  it  is  noon,  or  12  o'clock,  at  that 
place;  and  since  the  sun's  rays  pass  over  the  earth's  sur- 
face from  east  to  westy  it  is  after  noon  at  all  places  cast  of 
this  meridian,  and  before  noon  at  all  places  west  of  it. 
When  it  is  noon  at  Cincinnati,  it  is  after  noon  at  New 
York,  and  before  noon  at  St.  Louis. 

If  24  clocks  were  placed 
15°  apart  on  any  parallel  of 
latitude  between  the  polar 
circles,  the  difference  in  time 
between  any  two  consecutive 
clocks  would  be  one  hour; 
and  the  24  clocks  would  to- 
gether represent  every  hour 
of  the  day.  The  figures  in 
the  diagram  represent  the 
location  of  the  clocks  (15° 

apart),  and  also  the  hour  of  the  day  corresponding  to  noon  on  the 
meridian. 

MENTAL   PROBLEMS. 

1.  The  earth  rotates  on  its  axis  once  every  24  hours: 
what  part  of  a  rotation  does  it  make  in  1  hour? 

2.  How  many  degrees  of  the  earth's  surface  pass  under 
the  sun's  rays  in  24  hours?     In  1  hour? 


132  COMPLETE  ARITHMETIC. 

3.  How  many  degrees  of  longitude  make  a  difference  of 
1  hour  in  time? 

4.  When  it  is  noon  at  Washington,  what  is  the  hour 
of  day  15°  east  of  Washington?    15°  west  of  Washington? 

5.  When  it  is  6  o'clock  at  Boston,  what  is  the  hour  of 
day  30°  east  of  Boston?     45°  west  of  Boston? 

6.  If  15°  of  longitude  give  a  difference  of  1  hour  in 
time,  how  much  longitude  will  give  a  difference  of  1  min- 
ute in  time? 

7.  When  it  is  4  o'clock  at  Cincinnati,  what  is  the  time 
15'  east  of  Cincinnati?     45'  west  of  Cincinnati  ? 

8.  When  it  is  9  o'clock  at  Chicago,  what  is  the  time 
15°  15'  east  of  Chicago?     15°  45'  west  of  Chicago? 

9.  If  15'  difference  in  longitude  gives  a  difference  of  1 
minute  in  time,  what  difference  in  longitude  will  give  a 
difference  of  1  second  in  time  ? 

10.  What  difference  in  longitude  gives  a  difference  of  1 
hour  in  time  ?     1  minute  ?     1  second  ? 

11.  The  difference  in  time  between  two  cities  is  2  hours: 
what  is  the  difference  in  their  longitude?  Which  has  the 
earlier  time  ? 

12.  The  difference  in  time  between  New  York  and  St.  Louis 
is  1  h.  2^  min. :  what  is  the  difference  in  their  longitude  ? 

13.  A  gentleman  left  Boston  and  traveled  until  his  watch 
was  1  h.  3  min.  too  slow :  how  far  had  he  traveled,  and  in 
which  direction? 

14.  Two  captains  observed  an  eclipse  of  the  moon,  one 
seeing  it  at  9  P.  M.  and  the  other  at  11^  P.  M. :  what  was 
the  difference  in  their  longitude? 

WRITTEN"  PROBLEMS. 

15.  The  difference  in  longitude  between  two  places  is  31° 
45'  30" :  what  is  the  difference  of  time  ? 

Process. 
15  )  31°         45^         30^^         Divide  by  15  as  in  compound  division. 
2  li.    7  rain.    2  sec. 


LONGITUDE  AND  TIME.  133 

16.  The  difference  in  longitude  between  two  cities  is  5° 
31':   what  is  the  difference  in  time? 

17.  The  longitude  of  Cincinnati  is  84°  26'  W.,  and  San 
Francisco  is  122°  26'  15"  W. :  when  it  is  noon  at  Cincin- 
nati what  is  the  time  at  San  Francisco? 

18.  Philadelphia  is  75°  10'  W.  :  when  it  is  noon  at  San 
Francisco  what  is  the  time  at  Philadelphia? 

19.  Boston  is  71°  4'  9"  W.:  when  it  is  7  P.  M.  at  Boston 
what  is  the  time  at  Cincinnati?     At  San  Francisco? 

20.  Berlin  is  13°  23'  53"  E. :  when  it  is  noon  at  Boston 
what  is  the  time  at  Berlin  ? 

21.  The  difference  in  time  between  two  cities  is  1  h. 
35  min.  12  sec:   what  is  their  difference  in  longitude? 

Pkocess. 
1  h.  35  min.  12  sec.  Multiply  by  15  as  in  compound  multipli- 

c ^'J  cation. 

23°        48^       0^^  Ans, 

22.  The  difference  in  time  in  the  observations  of  an 
eclipse,  on  two  vessels  at  sea,  is  2  h.  15  min.  10  sec. :  what 
is  their  difference  in  longitude? 

23.  An  eclipse  was  observed  at  New  York,  74°  W.,  at 
9.30  P.  M.,  and  the  time  of  its  observation  on  a  vessel  in 
the  Atlantic  Ocean,  was  11.45  P.  M. :  what  was  the  longi- 
tude of  the  vessel? 

24.  The  difference  in  time  between  the  chronometers  of 
two  observatories  is  45  min.  30  sec,  and  the  longitude  of 
the  observatory  having  the  earlier  or  faster  time  is  85° 
40'  W. :  what  is  the  longitude  of  the  other  observatory  ? 

25.  The  distance  from  Boston  to  Chicago  is  about  890 
miles,  and  a  degree  of  longitude  at  Boston  contains  about 
52  miles:  when  it  is  noon  at  Boston  what  is  the  time  at 
Chicago  ? 

26.  The  distance  from  Washington  to  St.  Louis  is  about 
690  miles,  and  a  degree  of  longitude  at  Washington  con- 
tains about  53  miles :  when  it  is  9  o'clock  at  St.  Louis  what 
is  the  time  at  Washington  ? 


134  COMPLETE  AKITHMETIC. 

27.  The  difference  in  the  longitude  of  two  vessels,  at  the 
time  of  the  observation  of  an  eclipse,  Avas  25°  30':  what 
was  their  difference  in  time? 

28.  How  much  earlier  does  the  sun  rise  at  Baltimore, 
which  is  76°  37'  W.,  than  at  St.  Louis? 

29.  How  much  later  does  the  sun  set  at  Chicago,  which 
is  87°  35'  ^y.,  than  at  Boston? 

30.  How  much  later  does  the  sun  set  at  San  Francisco 
than  at  Cincinnati  ? 

31.  How  much  earlier  does  the  sun  rise  at  New  York 
than  at  San  Francisco? 

32.  How  much  later  does  the  sun  set  at  Boston  than  at 
Berlin? 

TABLE  AND  RULES. 

200.  Table  :   15°  difference  in  long,  gives  a  difference 

-  of  1  h.  in  time. 

15'   difference   in    long,   gives  a  difference 

of  1  m.  in  time. 
15"  difference   in   long,  gives  a  difference 

of  1  sec.  in  time. 

201.  EuLES. — 1.  To  find  the  difference  in  time  corre- 
sponding to  any  difference  in  longitude.  Divide  the  difference 
in  longitude,  expressed  in  degrees,  minutes,  and  seconds,  by  15, 
and  the  respective  quotients  will  be  hours,  mimdes,  and  seconds 
of  time. 

2.  To  find  the  difference  in  time  corresponding  to  any 
difference  in  longitude,  Multiply  the  difference  in  time,  ex- 
pressed in  hours,  minutes,  and  seconds,  by  15,  and  the  respect- 
ive products  ivill  be  degrees,  minutes,  and  seconds  of  longitude. 

3.  To  find  the  time  at  one  place  when  the  time  at 
another  place  and  their  difference  of  time  are  known,  When 
the  second  place  is  east  of  the  first,  add  their  difference  of 
time;  when  it  is  west  of  the  first,  subtract  their  difference 
of  time. 


PERCENTAGE.  135 

SECTION     XIV. 

PERCENTAGE. 

NOTATION  AND  DEFINITIONS. 

202.  One  per  cent,  of  a  number  is  one  hundredth  of  it; 
two  per  cent,  is  two  hundredths ;  and,  generally,  any  per 
cent,  of  a  number  is  so  many  hundredths  of  it. 

1.  How  many  hundredths  of  a  number   is  4  per  cent, 
of  it?     7  per  cent,  of  it?     15  per  cent,  of  it? 

2.  How  many  hundredths  of  a  number  is  3J  per  cent 
of  it?     121  per  cent.?     33-J-  per  cent.  ? 

3.  How  many  hundredths  of  a  number  is  \  of  one  per 
cent,  of  it  ?     f  of  one  per  cent.  ?     f  of  one  per  cent.  ? 

4.  How  many  hundredths  of  a  number  is  115  per  cent, 
of  it?     135  per  cent.?     180  per  cent.? 

5.  What  per  cent,  of  a  number  is  .05  of  it?     .09  of  it? 
.15  of  it?     .35  of  it? 

6.  What  per  cent,  of  a  number  is  .03^^  of  it?     .33 J  of 
it?     .OOfofit?     .001  of  it? 

7.  What  per  cent,  of  a  number  is  j-J^  of  it?    HJ  of  it? 
1.25  of  it?     1.65  of  it? 

8.  How  many  hundredths  of  a  number  is  45  %   of  it? 
110%?     125%?     170%?     215%? 

Note. — Tlie  character  %  is  often  nsed  instead  of  the  words  "  per 
cent."     Thus,  15  %  denotes  15  per  cent, 

9.  What  fractional  part  of  a  number  is   10  %   of  it? 

20%?     25%?     e50%? 

Solution.—  50  %  of  a  number  is  y\%  or  h  of  it. 

10.   What  fractional  part  of  a  number  is  12^  %  of  it  ? 

16|%?    331%?     (]G|%? 


136  COMPLETE   ARITHMETIC. 

11.  What  fractional  part  of  100  %  is  25  %  ?  50  %  ? 
75%?     66t%? 

WRITTEN   EXERCISES. 

12.  Express  decimally  1  % ;  3%;  6%;  7%;  8%;  9%. 

13.  Express  decimally  15%;  20%;  25%;  33%;  45%. 

14.  Express  decimally  112%;  125%;   150%;  220%. 

15.  Express  decimally  6i  %  ;  3^  %  ;  12^%;  24|%. 

Note. — The  fractional  part  of  one  per  cent,  may  be  expressed  deci- 
mally, as  thousandths,  ten-thousandths,  etc.  Thus,  6i  4  =  .06k  or 
.0625. 

16.  Express  decimally  7f  %  ;   16|%;   10^%;  30-J-%. 

17.  Express  decimally  1%;  |%;'|%;  f%;  ^%. 

18.  Express  decimally  5  %  ;  t\  %  ;  7 A  %  ;  i%;   20J-%. 

DEFINITIONS. 

203.  Any  JPer  Cent,  of  a  number  or  quantity  is  so 
many  hundredths  of  it. 

Note. — The  term  per  cent,  is  a  contraction  of  tlie  Latin  per  centum, 
wliieh  means  by  the  hundred. 

204.  The  Bate  JPer  Cent,  is  the  fraction  denoting 
the  number  of  hundredths  taken. 

The  rate  per  cent,  is  expressed  numerically  either  as  a  common 
fraction  or  as  a  decimal. 

205.  The  Mate  is  the  number  of  hundredths. 

Note. — In  this  work,  the  terms  Per  Cent.,  Rate  Per  Cent.,  and  Rate, 
are  not  used  as  synonymous.  In  the  statement,  "12  is  6  per  cent,  of 
200,"  12  is  considered  the  per  cent. ;  y^^,  or  .06,  as  the  rate  per  cent.  ; 
and  6  as  the  rate.  The  rate  is  the  numerator  of  the  fraction  denoting 
the  rate  per  cent. 

206.  The  character  %  is  called  the  Per  Cent  Sign,  and 
is  read  per  cent. 

207.  JPercentage  embraces  all  numerical  operations 
in  which  one  hundred  is  the  basis  of  computation. 


PERCENTAGE.  137 


THE  FOUR  CASES  OF  PERCENTAGE. 

208.  Four  numbers  are  considered  in  percentage,  and 
such  is  the  relation  between  them  that,  if  any  two  of  them 
are  given, the  other  two  may  be  found. 

These  four  numbers  are : 

1.  The  I^ase^  or  the  number  of  which  the  per  cent,  is 
found. 

2.  The  Hate  Per  Cent,,  or  the  fraction  denoting  the 
number  of  hundredths  of  the  base  taken. 

3.  The  Per  Cent,,  or  the  part  of  the  base  correspond- 
ing to  the  rate  per  cent.     It  is  also  called  the  Percentage. 

4.  The  Amount  or  Uifferenee,  or  the  number  ob- 
tained by  adding  the  per  cent,  to,  or  subtracting  it  from, 
the  base. 

Case  I. 

The   Base   and   tlie   Rate  Pei*  Cent,  given,  to  find 
the   Fei-   Cent. 

1.  How  much  is  5%  of  800? 

Solutions.— 1.  Since  5  %  =^  yf ^j,  5  %  of  800  =  y^o  "^  80f>,  ^vhich 
is  40.     Or, 

2.  1  %  of  800  =  y^^  of  800,  which  is  8 ;  and  5  %  of  800  ------  5  times 

8,  which  is  40. 

2.  What  is  6%  of  1200?     8%  of  250?     9%  of  4000? 

3.  What  is  5  %  of  $300?     8  %  of  $450?     12  %  of  $500? 

4.  What  is  6%  of  500  miles?  10%  of  250  miles?  \b% 
of  600  miles  ?     20  %  of  300  miles  ? 

WKITTEN  PROBLEMS. 

5.  What  is  8%  of  674.50? 

1st  Process.  2d  Process. 

$674.50  $6,745  --  1  % 

.08  8 


$53.96  00  $53,960  =  8  % 

Note. — Let  the  pupil  use  cme  method  until  he  is  ftimiliar  with  it, 
(\Ar.— 12. 


138  COMPLETE  ARITHMETIC. 


What  is 

6. 

5%  of  245? 

16. 

\%  of  $540? 

7. 

9%  of  360? 

17. 

tV%  of  $4000? 

8. 

15%  of  1200? 

18. 

f  %  of  21700  ft.? 

9: 

25%  of  37.5? 

19. 

1%  of  $48.50? 

10. 

33%  of  $150? 

20. 

331%  of  965  days? 

11. 

8%  of  $37.50? 

21. 

16|%  of  $.54? 

12. 

3%  of  $180.25? 

22. 

15%  off?     Of^^? 

13. 

H  %  of  1050  lb.  ? 

23. 

66|%of,^?     Of2|-? 

14. 

21%  of  60.8  1b.? 

24. 

12%  of  .25?     Of  .45? 

15. 

121-  ^  of  560  days  ? 

25. 

61%  of  50?     Of  .75? 

26.  What  is  the  difference  between  33%  and  25  J- %  of 
480  miles  ? 

27.  If  70  %  of  a  certain  ore  is  iron,  how  much  iron  is 
there  in  3740  pounds  of  ore? 

28.  If  20  %  of  air-dried  wood  is  water,  how  much  water 
is  there  in  143J  tons  of  wood? 

29.  A  man  receives  $1650  a  year,  and  his  expenses  are 
87^%  of  his  income:   how  much  has  he  left? 

30.  A  grain  dealer  owning  58500  bushels  of  wheat, 
shipped  371  %  of  it  by  a  steamer,  33^  %  of  it  by  a 
schooner,  and  the  rest  of  it  by  railroad :  how  many  bushels 
did  he  ship  by  each? 

FORMI^LAS  AND  RULES. 

209.  Formulas. — 1.  Per  cent.   ==^  base  X  rate  per  cent. 

2.  Amount     =  base  -\-  per  cent. 

3.  Difference  ^=  base — percent. 

210.  Rules. — To  find  a  given  per  cent,  of  any  number, 

1.  Multiply  the  given  number  by  the  rate  per  cent,  expressed 
decimally.     Or, 

2.  Eemx)ve  the  decimal  point  two  places  to  the  left,  and  mul- 
tiply the  result  by  the  rate. 

Note. — When  the  rate  is  an  aliquot  part  of  100,  tlie  per  cent,  may 
be  found  by  taking  the  same  aliquot  part  of  the  base.  Thus,  33]  % 
of  $48  —  ]  of  $48.  The  process  in  each  of  the  succeeding  cases  may 
be  shortened  by  using  the  fraction  denoting  the  alicpiot  part. 


PERCENTAGE.  139 


Case  11. 

a^h.e   Base   tiiad   the  Per  Cent,  given, to  find   tlie 
Rate  r*er  Cent. 

1.  What  per  cent,  of  16  is  4? 

Solutions.— 1.  1  is  j\  of  16,  and  4  is  j%  or  .25  (Art.  121).    Hence, 
4  is  25%  of  16.     Or, 

2.  1%  of  16  is  .16,  and  4  is  as  many  per  cent,  of  16  as  .16  is  con- 
tained times  in  4.00,  which  is  25. 

2.  What  per  cent,  of  $50  are  $5?     $20? 

3.  What  per  cent,  of  $300  are  $12?     $30? 

4.  What  per  cent,  of  150  lb.  are  75  lb.  ?     50  lb.? 

5.  What  per  cent,  of  250  ft.  arc  50  ft.?     100  ft.? 

WRITTEN  PROBLEMS. 

6.  AVhat  per  cent,  of  62.5  is  16? 

1st  Process.  2d  Process. 

16  ^  62.5  =  .24  Ifc  of  62.5  --=  .625 

.24  =:  24%,  Ans.  16  ^  .625  =  24,  Rate. 

Note.— The  quotient  obtained  by  the  first  procesn  is  the  rate  pa- 
cent.,  and  the  quotient  obtained  by  the  second  process  is  the  rate. 

What  per  cent,  of 

7.  75  is  4.5? 

8.  125  is  25  ? 

9.  120  is  40  ? 

10.  $450  are  $90? 

11.  $192  are  $32? 

12.  $760  are  $19? 

13.  $1000  are  $5? 

14.  $6  are  45  cts.  ? 

23.  A  farmer  had  320  sheep  and  sold  48  of  them :  what 
per  cent,  of  the  flock  did  he  sell  ? 

24.  A  gold  ring  is  22  carats  fine :  what  per  cent,  of  it  is 
gold? 

25.  What  per  cent,  of  $45  is  16|%  of  $150? 


15. 

75  lb.  are  16.5  lb.? 

16. 

20  ft.  are  1.2  ft.? 

17. 

371  yd.  are  5  yd.  ? 

18. 

.75  is  .15? 

19. 

.60  is  .45? 

20. 

fisi? 

21. 

*isF 

22. 

2|  is  f  ? 

140  COMPLETE  ARITHMETIC. 

26.  A  regiment  of  750  men  lost  160  men  in  a  certain 
battle:   what  per  cent,  of  the  regiment  remained? 

27.  What  per  cent,  of  any  number  is  |  of  it?     |  of  it? 

l^ofit?        2^0    of   it? 

28.  AVhat  per  cent,  of  a  number  is  |  of  it  ?  2^-  of  it  ? 
Ilofit?     i  off  of  it? 

FORMULA  AND  RULES. 

211.  Formula. — Rate  %  =z  per  cent  -^  base. 

212.  KuLES. — To  find  what  per  cent,  one  number  is  of 
another, 

1.  Divide  the  number  which  is  the  per  cent,  by  the  base,  and 
the  quotient  expressed  in  hundredths  ivill  be  the  rate  per  cent. 
Or, 

2.  Divide  the  number  which  is  the  pter  cent,  by  one  per  cent, 
of  the  base,  ami  the  quotient  will  be  the  rate. 

Case  III. 

3?ei'  Certt.   tvncl  Rate  Fer  Cent,  giveia,  to  find  th.e 
JBtise. 

1.  45  is  15  %  of  what  number? 

Solutions.— 1.  If  15%,  or  .15,  of  a  number  is  45,  the  number 
equals  45  -f-  .15,  which  is  300.     Or, 

2.  If  45  is  15%  of  a  number,  1%  of  it  is  y^  ^f  45,  which  is  3,  and 
100%,  or  the  number,  is  100  times  3,  which  is  300. 

2.  320  is  16%  of  what  number? 

3.  7.2  pounds  are  12%  of  how  many  pounds? 

4.  Charles  is  15  years  old,  and  his  age  is  30%  of  his 
father's  age :  how"  old  is  his  father  ? 

5.  A  man's  expenses  are  $28  a  month,  which  is  70%  of 
his  wages :  how  much  does  he  earn  a  month  ? 

6.  In  a  certain  school  56  pupils  study  arithmetic,  which 
is  28  per  cent,  of  the  whole  number  of  pupils  in  school : 
how  many  pupils  in  the  school  ? 


PERCENTAGE.  141 

'WRITTEN   PROBLEMS. 

7.  A  man  owes  $4560,  which  is  30%  of  his  estate:  how 
much  is  his  estate? 

1st  Process.  2d  Process. 

$4560  -^  .30  =  $15200,  Ans.  $4560  ^  30  X  100  =  $15200, 

8.  256  is  35%  of  what  number? 

9.  1331  is  162  %  of  what  number? 

10.  1071  is  15%  of  what  number? 

11.  480  sheep  are  36%  of  how  many  sheep? 

12.  5280  pounds  are  66f  %  of  how  many  pounds  ? 

13.  $189.80  are  104%  of  what  sum  of  money? 

14.  $88.66  are  110%  of  what  sum  of  money? 

15.  The  number  of  pupils  in  daily  attendance  in  a  cer- 
tain school  is  420,  which  is  80  %  of  the  number  enrolled : 
how  many  pupils  are  enrolled  ? 

16.  The  number  of  youth  of  school  age  in  a  certain  city 
is  5220,  Avhich  is  36  %  of  the  number  of  inhabitants :  what 
is  the  population  of  the  city  ? 

17.  A  man  spent  60  %  of  his  money  for  a  suit  of  clothes, 
25  %  of  it  for  books,  and  had  $7.50  left:  how  much  money 
had  he  at  first? 

18.  A  man  invested  $5400  in  railroad  stock,  which  was 
37^%  of  his  property:  what  was  the  value  of  his  property? 

19.  In  a  storm,  a  ship's  crew  threw  overboard  250  bar- 
rels of  flour,  which  was  40  %  of  the  number  of  barrels  on 
board :   how  many  barrels  of  flour  were  left  on  board  ? 

20.  A  man  owning  60  %  of  a  factory,  sold  40  %  of  his 
share  for  $9600 :  at  this  rate,  what  was  the  value  of  the 
factory  ? 

21.  The  land  surface  of  the  earth  is  about  50000000 
sq.  miles,  which  is  331  %  of  the  water  surface :  what  is 
the  extent  of  the  water  surface? 

22.  The  population  of  a  certain  city  in  1860  was  64000, 
which  is  80%  of  the  population  in  1870:  what  was  the 
population  in  1870? 


142  COMPLETE   ARITHMETIC. 

FORMULA  AND  RULES. 

213.  Formula.— ^ase  =rz  ^er  cent  -^-  rate  per  cent 

214.  KuLES. — To  find  a  number  when  a  certain  per  cent, 
of  it  is  given, 

1.  Divide  the  number  whidi  is  tJie  per  cent,  by  the  rate  per 
cent,  expressed  decimally.     Or, 

2.  Divide  tJie  number  whidh  is  Uie  per  cent,  by  the  rate,  and 
multiply  the  quotient  by  100. 

Case  IV. 

The  Ainount  or  DifFereiice  and  the  Rate  I^er  Cent, 
given,   to   finci   the  Base. 

1.  216  is  8  %  more  than  what  number  ? 

Solutions. — 1.  If  216  is  8%  more  than  a  number,  216  is  108%  or 
1.08  of  it,  and  hence  the  number  equals  216  -^- 1.08,  which  is  200.    Or, 

2.  If  216  is  108^  of.  a  number,  1%  of  it  is  xk  of  216,  which  is  2, 
and  100%  is  100  times  2,  which  is  200. 

2.  318  is  6  %  more  than  what  number  ? 

3.  $480  is   20  ^  more  than  what  sum  of  money? 

4.  560  pounds  are  12%  more  than  how  many  pounds? 

5.  184  is  8  %  less  than  what  number  ? 

Suggestion.— If  184  is  8%  less  than  a  number,  184  is  100^  —  8%, 
or  92%  of  it. 

6.  285  is  5  %  less  than  what  number? 

7.  $356  are  11  %  less  than  how  many  dollars? 

8.  425  feet  are  15  %  less  than  how  many  feet? 

9.  A  horse  cost  $160,*  which  was  20%  less  than  the  cost 
of  a  carriage :   what  was  the  cost  of  the  carriage  ? 

10.  A  school  enrolls  230  boys,  which  is  15  %  more  than 
the  number  of  girls  enrolled:  hoAV  many  pupils  in  the 
school  ? 

WRITTEN  PROBLEMS. 

11.  A  farm  was  sold  for  $6390,  which  was  12^%  more 
than  it  cost :  what  was  the  cost  of  the  farm  ? 


PERCENTAGE.  143 

1st  Process.  2d  Process. 

100%  +12i%  =-112.1%  =1.125  $6390 -^112.5  =  $56.80  =  1% 

$6390  -^  1.125  =  $5680,  Ans.     $56.80  X 100  =  $5680,  Ans. 

12.  A  man's  expenses  are  $400  a  year,  which  is  31-^% 
less  than  his  income :  what  is  his  income  ? 

13.  276  is  15%  more  than  what  number? 

14.  What  number  increased  by  30  %  of  itself,  equals 
162.5? 

15.  What  number  diminished  by  16|%  of  itself,  equals 
2035.8? 

16.  A's  farm  contains  306  acres,  which  is  32  %  less  than 
B's :  how  many  acres  in  B's  farm  ? 

17.  When  gold  was  worth  25  %  more  than  currency,  what 
was  the  gold  value  of  $150  in  currency? 

18.  When  gold  was  worth  50  %  more  than  currency, 
what  was  the  value  in  gold  of  a  dollar  bill  ? 

19.  The  number  of  pupils  in  daily  attendance  at  a  school 
is  570,  which  is  5  %  less  than  the  number  enrolled :  how 
many  pupils  are  enrolled? 

20.  The  number  of  jiupils  enrolled  in  a  certain  town  is 
920,  which  is  15  %  more  than  the  average  number  of  pupils 
in  daily  attendance :  what  is  the  average  daily  attendance  ? 

21.  The  population  of  a  certain  city  in  1870  was  171527, 
which  is  18  %  more  than  its  population  in  1860 :  what  was 
the  population  in  1860? 

FORMULAS  AND  RULES. 

215.  Formulas.— 1.  Base  =  amount     -^  (1  +  rate  %). 

2.  Base  =^  difference  -f-  (1  —  rate  %  ). 

216.  Rules. — To  find  a  number  when  another  number 
is  given,  which  is  a  given  rate  per  cent.,  more  or  less, 

1.  Divide  the  given  number  by  1  j9^w8  or  minus  the  given 
rate  per  cent,  expressed  decimally.    Or, 

2.  Divide  the  given  number  by  100  plus  or  minus  the  given 
rate,  and  multiply  the  quotient  by  100. 


144  COMPLETE  ARITHMETIC. 


REVIEW  OF  THE  FOUR  CASES. 

217.  The  formulas  of  the  four  preceding  cases  of  per- 
centage are  here  presented  together  for  comparison: 

Case  I. — Per  cent.  =  base  X  rate  per  cent. 

Case  II. — Rate  per  cent.  =  per  cent.  ^-  base. 

Case  III. — Base  =  per  cent.  -=-  rate  per  cent. 

^         ^^^       _,  f  Amount     -f-  (1 4-  rate  per  cent). 

Case  IY.—Base={  ^.^  ).  .  ^  /^ 

\  Vifference  -^  (1  —  rate  per  cent). 

Note. — Tlie  two  formulas,  amount  =  hose  +  per  cent,  and  difference 
=^b(ise  —  per  cent.,  do  not  involve  the  operations  of  percentage,  but 
simply  the  adding  and  subtracting  of  numbers. 

MENTAL   PROBLEMS. 

1.  What  is  121%  of  640?     16f  %  of  860? 

2.  What  is  33i  %  of  672  ?     66|  %  of  321  ? 

3.  15  is  what  per  cent,  of  60?  ^Of  90? 

4.  16f  lb.  is  what  per  cent,  of  50  lb.  ?     Of  100  lb.  ? 

5.  25%  of  120  is  what  per  cent,  of  90? 

6.  33i%  of  150  is  what  per  cent,  of  250? 

7.  80  is  12|^%  of  what  number? 

8.  20%  of  105  is  25%  of  what  number? 

9.  33^  %  of  225  is  15  %  of  what  number? 

10.  45  is  what  per  cent,  of  75%  of  120? 

11.  360  is  20%  more  than  what  number  ? 

12.  60  is  33  J  %  more  than  what  number  ? 

13.  33|-%  of  240  is  33^%  less  than  what  number? 

14.  25%  of  280  is  16|%  more  than  what  number? 

15.  A  man  is  60  years  of  age,  and  20  %  of  his  age  is 
25  %  of  the  age  of  his  wife :  how  old  is  his  wife  ? 

WRITTEN  PROBLEMS, 

16.  The  population  of  a  certain  city  in  1860  was  63500, 
and  the  census  of  1870  shows  an  increase  of  17J  % :  what 
was  the  population  in  1870? 


rERCEN'TAGE.  145 

17.  A  farm  contains  480  acres,  of  which  30  %  is  meadow, 
25|-%  pasture,  16f  %  grain  land,  and  the  rest  w^oodland: 
how  many  acres  of  each  kind  of  land  in  the  farm  ? 

18.  A  merchant  failed  in  business  owing  $10500  and 
having  $6300  worth  of  property:  what  per  cent,  of  his 
indebtedness  can  he  pay? 

19.  A  clerk,  receiving  a  yearly  salary  of  $950,  pays  $275 
a  year  for  board,  $180  for  clothing,  and  $150  for  other 
expenses :   what  per  cent,  of  his  salary  is  left  ? 

20.  A  lady  pays  $280  a  year  for  board,  $175  a  year  for 
clothing  and  other  expenses,  and  lays  up  35  %  of  her  in- 
come: what  is  her  income? 

21.  A  man's  expenses  are  80%  of  his  income,  and  33  J  % 
of  his  income  equals  10  %  of  his  property,  which  is  valued 
at  $27000 :  what  are  his  expenses  ? 

22.  A  merchant  sold  a  stock  of  goods  for  $10811,  and 
gained  13-|  %  :  what  was  the  cost  of  the  goods  ? 

23.  A  cargo  of  damaged  corn  was  sold  at  auction  for 
$9450,  which  was  33^  %  less  than  cost :  what  was  the  cost 
of  the  corn? 

24.  An  orchard  contains  1200  trees,  of  which  45  %  are 
apple,  22  %  peach,  12|^  %  cherry,  and  the  rest  pear :  how 
many  trees  of  each  kind  in  the  orchard? 

25.  A  owns  42^%  of  a  factory  worth  $35000,  B  owns 
37  %  of  it,  and  C  owns  the  remainder :  what  is  the  value 
of  each  of  their  shares? 

26.  A  man  bequeathed  $7560  to  his  wife,  which  was 
Q2^%  of  the  sum  bequeathed  to  his  children,  and  the  sum 
bequeathed  to  his  wife  and  children  was  80  %  of  his  estate : 
what  was  the  value  of  the  estate? 

27.  The  population  of  a  city  in  1870  was  41064,  which 
was  16  %  more  than  in  1860,  and  the  population  in  1860 
was  6^  %  less  than  in  1865 :  what  was  the  population  in 
1865? 

28.  The  number  of  deaths  in  a  certaiji  city  in  1869  was 
1960,  which  was  equal  to  3^  %  of  the  population :  what  was 
the  population  ?  ... 

C.Ar.— 13. 


146  COMPLETE    ARITHMETIC. 


APPLICATIONS  OF  PERCENTAGE. 

218.  The  principal  applications  of  percentage  are  Profit 
and  Loss,  Commission  and  Brokerage,  Capital  and  Stocks, 
Insurance,  Taxes,  Customs,  Bankriiptcy,  Interest,  Discount, 
Exchange,  Equation  of  Payments,  and  Equation  of  Accounts. 

All  the  problems  are  solved  by  the  application  of  one  or 
more  of  the  four  cases  of  percentage. 

PROFIT  AND  LOSS. 

219.  The  Cost  of  an  article  is  the  price  paid  for  it,  or 
the  total  expense  incurred  in  producing  it. 

220.  The  Selling  JPrice  of  an  article  is  the  amount 
asked  or  received  for  it  by  the  seller. 

The  selling  price  of  the  seller  is  the  cost  to  the  buyer,  and  vice  versa. 

221.  When  an  article  is  sold  for  more  than  its  cost,  it  is 
said  to  be  sold  at  a  p?'q/i^  or  gain;  when  it  is  sold  for  less 
than  its  cost,  it  is  said  to  be  sold  at  a  loss  or  discount. 
Hence, 

222.  J^rofit  or  Grain  is  the  amount  which  the  selling 
price  of  an  article  exceeds  its  cost. 

223.  Loss  or  Discount  is  the  amount  which  the  sell- 
ing price  of  an  article  is  less  than  its  cost. 

Note. — The  terms  gain  and  loss  are  not  limited  to  business  trans- 
actions. When  any  quantity  undergoes  an  increase  or  decrease,  from 
any  cause,  there  is  a  gain  or  loss,  and  when  such  gain  or  loss  can  be 
expressed  in  hundredths,  it  may  be  computed  by  the  principles  of 
percentage. 

MENTAL   PROBLEMS. 

1.  A  merchant  bought  a  piece  of  cloth  for  $80,  and  sold 
it  at  25%  profit:   for  how  much  did  he  sell  it?     (Case  I.) 

2.  A  dealer  bought  hats  at  $5  apiece,  and  sold  them  at 
20  %  profit :  what  was  the  selling  price  ? 


PROFIT  AND  LOSS.  147 

3.  Hats,  costing  $5  apiece,  were  sold  at  a  loss  of  20  % : 
what  was  the  selling  price  ? 

4.  At  what  price  must  flour,  costing  $6  a  barrel,  be  sold 
to  gain  16|  %  ? 

5.  A  grocer  bought  sugar  at  12  cts.,  16  cts.,  and  18  ct^. 
a  pound:  for  how  much  must  each  kind  be  sold  to  gain 
20  %  ?     To  gain  25  %  ? 

6.  A  merchant  sells  broadcloth,  costing  $4,  for  $5  a 
yard:  what  per  cent,  does  he  gain?     (Case  IL) 

7.  When  broadcloth,  costing  $5  a  yard,  is  sold  for  ^4  a 
yard,  what  is  the  loss  per  cent.  ? 

8.  Teas  costing  $1.20  and  $1.50  a  pound,  are  sold 
respectively  at  $1.50  and  $1.80  a  pound:  what  is  the  gain 
per  cent.? 

9.  A  merchant  sold  velvet  at  a  profit  of  $2  a  yard,  and 
gained  20%:  how  much  did  it  cost?     {Case  III.) 

10.  A  dealer  sold  boots  at  $1.50  a  pair  less  than  cost, 
and  thereby  lost  33^%  :   what  did  they  cost? 

11.  A  grocer  sold  tea  at  30  cents  above  cost,  and  gained 
lQj%  :  what  was  the  cost  of  the  tea?  What  was  the  sell- 
ing price  ? 

12.  A  man  sold  a  horse  for  $90,  and  gained  20  %  :  what 
was  the  cost  of  the  horse  ?     {Case  IV.) 

13.  A  man  sold  a  horse  for  $80,  and  lost  20  %  :  what  was 
the  cost  of  the  horse? 

14.  Sold  butter  at  40  cts.  a  pound,  and  gained  25  %  : 
how  much  did  it  cost? 

15.  A  watch,  costing  $80,  was  sold  at  a  loss  of  10  %  :  for 
how  much  was  it  sold  ? 

16.  How  must  shoes,  costing  $2,  $2.50,  and  $3  a  pair,  be 
sold  respectively  to  gain  25  %  ?  How  must  each  kind  be 
sold  to  gain  30  %  ? 

17.  How  must  muslin  that  cost  10  cts.,  15  cts.,  and  18 
cts.  a  yard,  be  sold  to  gain  20  %  ? 

18.  Sold  tea  at  90  cts.  a  pound,  and  gained  20  %  :  what 
would  have  been  my  gain  per  cent,  had  I  sold  it  at  $1  a 
pound  ? 


148  COMPLETE  ARITHMETIC. 


AVRITTEN   PKOBIiEMS. 

19.  A  house  and  lot,  which  cost  $67o0,  were  sold  at  a 
gain  of  12-^-  %  :   for  how  much  were  they  sold  ? 

20.  Carriages,  costing  $165,  are  sold  at  18%  profit:  what 
is  the  gain  on  each  carriage? 

21.  A  man  paid  $4500  for  a  farm,  and  sold  it  for  $5400 : 
what  was  the  gain  per  cent.  ? 

22.  A  drover  bought  cattle  at  $65  a  head,  and  sold  them 
at  $84.50  a  head:  what  was  the  gain  per  cent.? 

23.  Carpeting,  costing  $1.75  a  yard,  is  sold  for  $2:  what 
is  the  gain  per  cent.  ? 

24.  A  drover  bought  horses  at  $130  a  head,  expended  $6 
each  in  taking  them  to  market,  and  then  sold  them  at 
$153.50  a  head:  what  was  the  gain  per  cent.? 

25.  A  cargo  of  wheat  costing  $16500,  being  damaged,  is 
sold  for  $13750 :   what  was  the  loss  per  cent.  ? 

26.  A  merchant  sold  a  lot  of  goods  at  12^%  profit,  and 
gained  $8160:   what  was  the  cost? 

27.  A  grocer  sold  21  barrels  of  apples  at  22  %  profit,  and 
gained  $45.10:   what  was  the  cost  per  barrel? 

28.  A  man  sold  a  watch  for  $180,  and  lost  16f  %  :  what 
was  the  cost  of  the  watch? 

29.  A  house  and  lot  were  sold  for  $7762.50,  at  a  gain  of 
15%:   what  was  the  cost? 

30.  A  dry  goods  firm  sold  $45000  worth  of  goods  in  a 
year,  |  of  the  goods  being  sold  at  20  %  profit,  |-  at  25  % 
profit,  and  the  rest  at  33|^  %  profit :  what  was  the  cost  of 
all  the  goods? 

31.  Sold  a  piece  of  carpeting  for  $240,  and  lost  20  %  : 
what  selling  price  would  have  given  a  gain  of  20  %  ? 

32.  A  merchant  sells  goods  at  retail  at  30  %  above  cost, 
and  at  wholesale  at  12%  less  than  the  retail  price:  what  is 
his  gain  per  cent,  on  goods  sold  at  wholesale? 

33.  How  must  cloth,  costing  $3.50  a  yard,  be  marked 
that  a  merchant  may  deduct  15  %  from  the  marked  price 
and  still  make  15%  profit? 


COMMISSION  AND  BROKERAGE.  149 

34.  A  merchant  marked  a  piece  of  silk  at  25%  above 
cost,  and  then  sold  it  at  20  %  less  than  the  marked  price : 
did  he  gain  or  lose,  and  how  much  ? 

FORMULAS  AND  RULES. 

224.  Formulas. — 1.  Gain  or  loss  =  cost  X  rate  %. 

.2.  Rate  per  cent.  =  gain  or  loss  -f-  cost 
3.  Cost  =  gain  or  loss  -f-  rate  %. 


4.  Cost  =  selling  price  -^  ^  i 


■rate%. 

225.  Rules.— 1.  To  find  the  gain  or  loss  when  the  cost 
and  rate  per  cent,  are  given,  Multiply  the  cost  by  the  rate  per 
cent,  expressed  decimally.     (Form.  1.) 

2.  To  find  the  rate  per  cent,  when  the  cost  and  the  gain 
or  loss  are  given.  Divide  the  gain  or  loss  by  the  cost,  and  the 
quotient  expressed  in  hundredths  ivill  be  the  rate  per  cent. 
(Form.  2.) 

3.  To  find  the  cost  when  the  gain  or  loss  and  the  rate 
per  cent,  are  given,  Divide  the  gain  or  loss  by  the  rate  per 
cent,  expressed  decimally.     (Form.  3.) 

4.  To  find  the  cost  when  the  selling  price  and  the  rate 
per  cent,  of  gain  or  loss  are  given,  Divide  the  selling  price 
by  1  plus  or  minus  the  rate  per  cent.     (Form.  4.) 

Note.— Let  the  pupil  review  the  above  problems,  solving  those  in 
wliich  tlie  rate  is  an  aliquot  part  of  100,  bv  using  the  fraction.  (Art. 
210,  Note.) 

COMMISSION  AND  BROKERAGE. 

226.  An  Agent  is  a  person  who  transacts  business  for 
another. 

227.  A  Factor  is  an  agent  who  buys  and  sells  goods 
intrusted  to  his  possession  and  control.  A  mercantile  factor 
is  also  called  a  Commission  Merchant. 

When  a  factor  lives  in  a  different  country  or  part  of  the  country 
from  his  employer,  he  is  called  a  Correspondent  or  Consignee.     The 


150  COMPLETE  ARITHMETIC. 

goods  shipped  or  consigned  to  a  Consignee  are  called  a  Consignment  ; 
and  the  sender  of  the  goods  is  called  a  Consignor. 

228.  A  JBroUer  is  a  person  who  buys  and  sells  gold, 
bills  of  exchange,  stocks,  bonds,  etc. ;  or  an  agent  who 
buys  and  sells  property  in  possession  of  others. 

229.  A  Collector  is  an  agent  who  collects  debts,  taxes, 
duties,  etc.     (Arts.  263,  267.) 

230.  Commission  is  an  allowance  made  to  a  factor  or 
other  agent,  for  the  transaction  of  business.  The  commis- 
sion allowed  a  broker  is  called  Brokerage. 

231.  Commission  is  computed  at  a  certain  per  cent,  of 
the  amount  of  property  bought  or  sold,  or  of  the  amount 
of  business  transacted.  The  rate  per  cent,  is  called  the  Rate 
of  Commission,  and  the  amount  of  business  transacted  is  the 
Base. 

The  rate  of  commission  varies  with  the  amount  and  nature  of  the 
business.     A  broker's  commission  is  usually  less  than  a  factor's. 

232.  The  Wet  Proceeds  of  a  sale  or  collection  are  the 
proceeds  less  the  commission  and  other  charges. 

MENTAL   PEOBLEMS. 

1.  An  auctioneer  sold  $300  worth  of  furniture,  and 
charged  a  commission  of  5  %  :  how  much  did  he  receive  ? 

2.  A  peddler  bought  $500  worth  of  rags,  at  a  commis- 
sion of  10%  :  what  was  his  commission? 

3.  An  agent  sold  $1200  worth  of  school  furniture,  at  a 
commission  of  16|  %  :  how  much  did  he  receive? 

4.  An  attorney  collected  bad  debts  to  the  amount  of 
$800,  and  charged  20  %  commission :  what  was  his  com- 
mission ? 

5.  A  society  paid  a  lad  $6  for  collecting  membership 
dues,  to  the  amount  of  $100:  what  rate  of  commission  did 
he  receive? 

6.  A  bookseller  received  $30  for  selling  $150  worth  of 
maps:  what  was  his  rate  of  commission? 


COMMISSION  AND  BROKERAGE.  151 

7.  A  real-estate  broker  received  $40  for  selling  a  house 
and  lot,  at  5%  commission:  for  how  much  was  the  prop- 
erty sold? 

8.  An  attorney  received  $60  for  collecting  a  note,  at 
10%  commission:  what  was  the  amount  collected? 

9.  An  agent  received  $108,  with  which  to  buy  peaches, 
after  deducting  his  commission  at  8  %  :  how  much  did  he 
expend  for  peaches? 

10.  A  factor  received  $309,  with  which  to  buy  flour, 
after  deducting  his  commission,  at  3  % :  what  was  the  cost 
of  the  flour? 

11.  A  lawyer  collected  a  bill  at  25  %  commission,  and 
remitted  $7.50  as  net  proceeds:  what  was  the  amount  col- 
lected?    What  was  the  lawyer's  commission? 

12.  A  bookseller  sold  a  lot  of  books  on  commission,  at 
20%,  and  remitted  $160  as  net  proceeds:  for  how  much 
were  the  books  sold? 

WRITTEN  PROBLEMS. 

13.  A  commission  merchant  sold  540  barrels  of  flour,  at 
$6.37^  a  barrel:  what  was  his  commission  at  3  %  ? 

14.  A  real-estate  broker  sold  325  acres  of  land,  at  $24.50 
an  acre,  and  charged  a  commission  of  2^  %  :  what  was  his 
commission? 

15.  An  auctioneer  sold  $5160.50  worth  of  dry  goods,  and 
$715.25  worth   of  furniture:   what   was  his  commission,  at 

16.  A  lawyer  collected  65  %  of  a  note  of  $950,  and 
charged  6^%  commission:  what  w^as  his  commission? 
What  was  the  amount  paid  over? 

17.  A   factor  in  New  Orleans   purchased  $75000  worth, 
of  cotton  for  a  Lowell  manufacturer,  at  If  %  commission : 
what  was  his  bill  for  cotton  and  commission? 

18.  An  architect  charged  ^  %  for  plans  and  specifica- 
tions, and  If  %  for  superintending  the  erection  of  a  build- 
ing, costing  $120000:  how  much  was  his  fee? 


152  COMPLETE  ARITHMETIC. 

19.  An  agent  furnished  a  school-house  for  $4500,  and 
received  $540  commission:  what  Avas  the  rate? 

20.  An  attorney  charged  $75  for  collecting  rents  to  the 
amount  of  $1125:   what  was  the  rate  of  commission? 

21.  A  commission  merchant  charged  2|-  %  for  buying 
produce,  and  his  commission  was  $750 :  how  much  produce 
did  he  purchase? 

22.  A  w^ool  agent  received  5  %  for  buying  wool,  and  his 
commission  was  $208.50:   how  much  wool  did  he  buy? 

23.  My  agent  has  bought  3300  barrels  of  apples,  at 
$1.75  a  barrel,  and  I  allow  him  3%  commission:  how 
much  money  must  I  remit  to  pay  both  the  cost  of  the 
apples  and  the  commission? 

24.  A  Boston  merchant  sent  his  factor  in  Cincinnati 
$3539.20,  to  be  invested  in  bacon,  after  deducting  his  com- 
mission at  2  %  :  how  much  did  he  expend  for  bacon,  and 
what  was  his  commission? 

25.  A  cotton  broker  in  Charleston  received  $11774,  with 
which  to  purchase  cotton,  after  deducting  his  commission  of 
1|-  %  :  how  much  did  he  expend  for  cotton,  and  what  was 
his  commission? 

26.  A  merchant  paid  a  broker  f  %  for  a  draft  of  $1280 
on  New  York :  how  much  was  the  brokerage  ? 

27.  A  broker  bought  $15600  worth  of  stocks,  and  charged 
|-  %  :  what  was  his  fee  ? 

28.  A  real  estate  broker  sold  a  section  of  land  (640  A.) 
at  $7.50  an  acre,  and  invested  the  proceeds  in  railroad  stock, 
receiving  1^  %  for  selling  the  land  and  J  %  for  buying  the 
stock  :  what  was  his  brokerage  ? 

29.  What  will  be  the  total  cost  of  750  yards  of  carpeting, 
if  a  merchant  pays  2^  %  commission  for  purchasing,  \  % 
for  a  draft  covering  cost  and  agent's  commission,  and  $12.50 
for  freight? 

30.  A  grain  dealer  in  Chicago  received  $5000  with  direc- 
tions to  purchase  wheat,  at  $1.10  a  bushel,  after  deducting 
his  commission  at  2^  %  :  how  many  bushels  of  wheat  did 
he  purchase? 


COMMISSION  AND  BROKERAGE.  153 

31.  An  agent  sold  45  sewing  machines  at  $75  apiece, 
and  9  at  $125  apiece,  and,  deducting  his  commission,  re- 
mitted $3375  to  the  manufacturer  as  proceeds:  what  was 
his  rate  of  commission  ? 

32.  A  factor  sold  $15000  worth  of  goods,  at  10  %  com- 
mission, and  invested  the  proceeds  in  cotton,  first  deducting 
5  %  commission  for  buying :  how  much  money  did  he  invest 
in  cotton? 

33.  Smith  &  Jones  sell  for  C.  Bell  &  Co.  3040  pounds 
of  butter,  at  22  cts.  a  pound,  and  10560  pounds  of  cheese, 
at  15  cts.  a  pound,  and  invest  the  proceeds  in  dry  goods, 
first  deducting  their  commission  of  5  %  for  selling  and  3  % 
for  buying  :  how  much  did  they  invest  in  dry  goods  ?  What 
was  their  entire  commission  ? 

34.  A  commission  merchant  sold  1300  barrels  of  flour, 
at  $5.75  a  barrel,  receiving  a  commission  of  3J^,  and  in- 
vested the  net  proceeds  in  coffee,  at  28  cts.  a  pound,  first 
deducting  2  %  commission :  how  many  pounds  of  coffee  did 
he  purchase?     What  was  his  entire  commission? 

FORMULAS  AND  RULES. 

233.  Formulas.— 1.   Com.  or  hroh.  =  base  X  rate  %. 

2.  Rate  %  =-z  com.  or  hroh.  -r-  hase. 

3.  Bai^e  =  com.  or  hroh.  -^  rate  %. 

4.  Base  =  {base  -[-  com.  m-  broh.)  -r-  (1  -{- 
rate  %.) 

234.  Rules. — 1.  To  find  commission  or  brokerage,  Mtd- 
tiply  the  sum  of  money  denoting  the  amount  of  b^mness  trans- 
acted, by  the  rate  per  cent,  expressed  decimally.     (Form.  1.) 

2.  To  find  the  sum  to  be  invested  Avhen  the  amount  given 
includes  both  the  sum  to  be  invested  and  the  commission 
or  brokerage.  Divide  the  given  amount  by  1  plus  the  rate  per 
cent.,  and  the  quotioit  luill  be  the  sum.  to  be  invested.    (Form.  4.) 

Note. — These  two  rules  cover  all  tlie  ordinary  business  transac- 
tions in  commission  or  brokerage,  but  the  pupil  should  be  required 
to  form  rules  embodying  each  of  the  four  formulas. 


154 


COMPLETE  ARITHMETIC. 


CAPITAL  AND  STOCK. 

235.   CajHtal  is  property  invested  in  trade,  manufac- 
tures, or  other  business. 

2Z6.TneI>ar  Value 

of  capital  is  its  original 
or  nominal  value. 
The  Market  Value 

of  capital  is  its  real  value, 
or  the  sum  for  which  it 
will  sell. 

When  the  market  value  of 
capital  equals  its  par  value, 
it  is  said  to  be  at  par ;  when 
the  market  value  is  more 
than  the  par  value,  it  is  above 
par,  or  at  a  premium;  when 
the  market  value  is  less  than 
the  par  value,  it  is  below  paj-, 
^jte>  ^^^  1-m— ^;s-T-^"-  or  at  a  discount. 

237.  Premium  is  the  amount  which  the  market  value 
of  capital  exceeds  its  par  value. 

238.  Discount  is  the  amount  which  the  market  value 
of  capital  is  less  than  its  par  value. 

239.  Premium  and  discount  are  computed  at  a  given  per 
cent,  of  the  par  value.  The  rate  per  cent,  is  called  the 
Rate  of  Premium,  or  the  Hate  of  Dhcount. 

240.  A  Com^Hiny  is  an  association  of  persons  united 
for  the  transaction  of  business. 

The  association  of  several  persons  in  business  as  partners,  bound 
by  articles  of  agreement,  is  called  a  Partnership,  and  the  company  is 
commonly  called  a  Firm  or  Hou?e.     (Art.  380.) 

241.  An  Incori^orated  Company  is  a  company 
organized  and  regulated  by  law.  It  is  called  a  Corporation, 
and  the  law  regulating  it  is  called  a  Charter. 


CAPITAL  AND  STOCK.  155 

242.  The  capital  of  a  corporation  is  called  Stock,  and  is 
divided  into  equal  parts,  usually  $100  each,  called  Shares. 
The  owners  of  these  shares  are  called  Stockholders. 

243.  Certificates  of  Stock  are  official  statements  of 
the  size  and  number  of  shares  owned  by  each  stockholder. 
They  are  called  Scrij),  and  are  bought  and  sold  like  other 
property. 

Stocks  are  at  par,  above  par,  or  below  par,  according  as  their  mar- 
ket value  equals,  exceeds,  or  is  less  than  their  par  value  or  face. 

The  market  value  of  stocks  is  quoted  at  a  certain  per  cent,  of  the 
par  value.  Stocks  quoted  at  108  are  worth  108  %  of  their  face,  that 
is,  are  8%  above  par;  stocks  quoted  at  92  are  worth  92%  of  their  face, 
that  is,  are  8%  below  par. 

The  business  of  buying  and  selling  stocks  is  called  Stock  Jobbing, 
and  persons  engaged  in  such  business  are  called  Stock  Jobbers,  or  Stock 
Brokers. 

244.  The  Gross  JEarninf/s  of  a  company  are  the 
total  receipts  from  its  business;  and  the  Net  Earnings  are 
the  net  profits,  found  by  deducting  all  expenses  and  losses 
from  the  gross  earnings. 

245.  A  Dividend  is  the  part  of  the  earnings  of  a 
company  distributed  among  the  stockholders. 

Dividends  are  usually  declared  annually  or  semi-annually,  and 
they  are  computed  as  a  per  cent,  of  the  piir  value  of  the  stock.  The 
rate  per  cent,  is  called  tlie  Rate  of  Dividend. 

246.  An  Assessment  is  a  sum  levied  upon  the  stock- 
holders to  meet  the  losses  or  expenses  of  the  business. 

The  business  of  incorporated  companies  is  usually  managed  by 
directors,  who  are  elected  by  the  stockholders,  each  being  entitled  to 
as  many  votes  as  he  owns  shares. 

Note. — When  a  business  corporation  wishes  to  raise  money  in  ad- 
dition to  that  derived  from  its  capital  stock,  it  issues  notes  or  bonds, 
payable  at  a  specified  time  with  interest,  and  secured  by  mortgage  on 
the  property  of  the  corporation.  These  notes  are  called  Mortgage 
Bonds,  and  their  owners  are  called  Bondholders.  These  bonds  are 
negotiable  and  are  called  Stocks  (Art,  330),  but  they  should  be  care- 
fully distinguished  from  Capital  Stock. 


156  COMPLETE  ARITHMETIC. 


MENTAL  PROBLEMS. 


1.  When  stock   is   6  %    premium,   what   is   the  market 
value  of  $1?     Of  $100? 

2.  When   stock  is   12  %    discount,  Avhat  is  the  market 
value  of  $1?     Of  $100? 

3.  How  much  will  5  shares  of  telegraph  stock  cost,  at 
4  %  premium  ?     At  4  %  discount  ? 

Note. —A  share  is  $100,  if  no  other  vakie  is  named. 

4.  How   is    stock  quoted   when    it   is   15%    premium? 
When  it  is  15  %  discount  ? 

5.  When  stock  is  quoted  at  107|^,  what  is  the  value  of 
$1?     Of  $100? 

6.  When  stock  is  quoted  at  87,  what  is  the  value  of  $1  ? 
Of  $100? 

7.  How  much  will  10  shares  of  mining  stock  cost  when 
quoted  at  104?     At  85?" 

8.  When  bank  stock  is  quoted  at  105,  how  many  shares 
can  be  bought  for  $525?     For  $840? 

9.  A  company  declares  a  dividend  of  3  %  :  how  much 
will  a  stockholder,  owning  15  shares,  receive? 

10.  A  manufacturing  company  made  an  assessment  of 
5%,  to  repair  damages  caused  by  a  freshet:  how  much 
must  a  stockholder,  owning  20  shares,  pay? 

"WKITTEN   PROBLEMS. 

11.  A  man  bought  75  shares  of  railroad  stock  at  1\% 
discount:  how  much  did  they  cost? 

12.  Bought  100  shares  of  Little  Miami  stock  at  109|-, 
and  sold  them  at  112^:  how  much  did  I  gain  in  the  trans- 
action ? 

13.  A  broker  bought  70  shares  of  insurance  stock  at 
6|-  %  premium,  and  sold  them  at  f  %  discount :  how  much 
did  he  lose  ? 


CAPITAL  AND  STOCK.  157 

14.  A  man  bought  52  shares  of  Illinois  Central  at  127  ; 
and  sold  36  shares  at  135,  and  the  rest  at  137|:  how  much 
did  he  gain  ? 

15.  A  man  exchanged  52  shares  of  railroad  stock  at  80, 
for  insurance  stock  at  104 :  how  many  shares  of  insurance 
stock  did  he  receive? 

16.  A  broker  bought  84  shares,  $50  each,  of  telegraph 
stock  at  94,  and  sold  them  at  lOOf:  how  much  did  he 
gain? 

17.  The  Cincinnati  Gas  Co.  declares  a  dividend  of 
16J  %  :  how  much  will  a  man  holding  36  shares  receive  ? 

18.  The  capital  of  an  insurance  company  is  $500000, 
and  it  declares  a  dividend  of  4|%  :  how  much  money  is 
distributed  among  the  stockholders? 

19.  A  company  with  a  capital  of  $125000  declares  a  div- 
idend of  4  % ,  with  $3500  surplus :  what  were  the  net  earn- 
ings of  the  company? 

Note. — The  surplus  is  a  part  of  the  net  earnings  set  apart  to  meet 
future  demands. 

20.  The  entire  capital  stock  of  the  railroads  in  Ohio  for 
1869  was  $106686116,  and  their  net  earnings  for  the  year 
were  $9051998:  what  was  the  average  rate  of  dividend? 

21.  The  net  earnings  of  a  gas  company  are  $22425,  and 
the  capital  stock  is  $215000:  what  rate  of  dividend  can  be 
declared,  no  surplus  being  reserved  ?  What  will  be  the  div- 
idend on  45  shares? 

22.  The  capital  of  a  mining  company  is  $450000;  the 
gross  receipts  are  $70680 ;  and  the  expenses  are  $40325 : 
what  rate  of  dividend  can  it  declare,  reserving  a  surplus 
of  $6505? 

23.  How  many  shares  of  bank  stock  at  4  ^  premium, 
can  be  bought  for  $8320  ? 

24.  How  much  railroad  stock,  at  12|  discount,  can  be 
bought  for  $8750  ? 

25.  When  N.  Y.  Central  is  quoted  at  95|,  how  much 
stock  can  be  bought  for  $6894,  brokerage  j  %^ 


158  COMPLETE  ARITHMETIC. 

26.  A  broker  bought  84  shares  of  coal  stock,  at  108 J,  re- 
ceived a  dividend  of  5.J  %,  and  then  sold  the  stock  for  106: 
how  much  did  he  gain  ? 

27.  A  broker  bought  stock  at  4  %  discount,  and,  selling 
the  same  at  5  %  premium,  gained  $450 :  how  many  shares 
did  he  purchase  ? 

28.  A  man  bought  Michigan  Central  at  120,  and  sold  at 
124 :   what  per  cent,  of  the  investment  did  he  gain  ? 

FORMULAS  AND  RULES. 

247.  Formulas. — 1 .  Dividend  or  assessment  =  stock  X  Tate  % . 

2.  B,ate  %  =  divid.  or  assess,  -f-  stock. 

3.  Stock  =  divid.  or  assess,  -f-  rate  % . 

4.  Prem.  or  dis.  =par  value  X  rate%. 

5.  Rate  %  ^^=prem.  or  dis.  -=-  par  value. 

6.  Par  valus=prem.  or  dis.  -~  rate  %. 

7.  Par  val.  =  market  val.  ^-  (1  dz  rate  %  )• 

8.  Market  val.  =  par  val.  X  (1  ih  rate%). 

9.  Market  val.  ^=par  val.  -\-prem.  or  —  dis. 

Note. — Formulae  4,  7,  and  8  cover  all  ordinary  transactions  in 
stock  jobbing.     Tlie  sign  dr ,  used  in  7  and  8,  is  read  plus  or  minus. 

248.  KuLES. — 1.  To  find  the  dividend  or  assessment  when 
the  stock  and  rate  per  cent,  are  given,  Midtiphj  the  stock  by 
the  rate  per  cent,  expressed  decimally.     (Form.  1.) 

2.  To  find  the  rate  per  cent,  when  the  dividend  or  assess- 
ment and  total  stock  are  given.  Divide  the  dividend  or  assess- 
ment by  the  amount  of  stock,  and  the  quotient,  expressed  in 
hundredths,  will  he  the  rate  per  cent.     (Form.  2.) 

3.  To  find  the  stock  when  the  dividend  or  assessment  and 
the  rate  per  cent,  are  given.  Divide  the  dividend  or  assessment 
by  the  rate  per  cent,  expressed  decimally.     (Form.  3.) 

4.  To  find  the  premium  or  discount  on  a  given  amount 
of  stock,  Midtiply  the  amount  of  stock  by  the  rate  per  cent,  ex- 
pressed decimally.     (Form.  4.) 


INSURANCE. 


Ud 


5.  To  find  the  cost  or  market  value  of  a  given  amount 
of  stock,  Multiply  the  amount  of  stock  (1)  by  1  plus  or  minus 
the  rate  per  cent.  (Form.  8)  ;  or  (2)  by  the  quoted  price  ex- 
pressed as  hundredUis. 

Note.  -The  cost  may  also  be  found  by  adding  the  premium  to,  or 
subtracting  the  discount  from,  the  par  value.     (Form.  9.) 

6.  To  find  the  amount  of  stock  which  can  be  bought  for 
a  given  amount  of  money.  Divide  the  amount  of  money  to  be 
invested  (1)  by  1  plus  or  minus  the  rate  per  cent.  (Form.  7) ; 
or  (2)  by  the  quoted  price  expressed  as  hundredths. 

Note. — When  brokerage  is  paid,  the  rate  of  brokerage  must  be 
subtracted  from  the  quoted  price  before  dividing. 


INSURANCE. 

249.  Insurance  is  a  guaranteed  indemnity  for  loss. 

250.  Fire  Insur- 
ance is  a  guaranteed  in- 
demnity for  loss  of  prop- 
erty by  fire. 

251.  Marine  In- 
surance is  a  guaran- 
teed indemnity  for  loss 
of  property  while  trans- 
ported by  water. 

Insurance  on  the  property 
transported  is  called  Cargo  In- 
surance; that  on  the  vessel  i-s 
called  Hull  Insurance, 

252.  Life  Insurance  is  a  guaranteed  indemnity  for 
loss  of  life. 

Health  Insurance  guarantees  the  insured  a  certain  sum  of  monev 
if  sick;  and  Accident  Insurance  pledges  a  like  indemnity  if  the  in- 
sured is  injured  by  accident. 


160  COMPLETE  ARITHMETIC. 

253.  The  I*olicy  is  the  written  contract  between  the 
insurer  and  the  insured. 

The  insurer  is  called  an  Underwriter,  and  the  insured  a  Policy 
Holder. 

254.  The  J^reniiuiU  is  the  sum  paid  by  the  insured 
to  obtain  the  insurance.  It  is  a  specified  per  cent,  of  the 
amount  insured. 

The  act  of  insuring  is  called  taking  a  risk.  When  property  is  in- 
sured, the  valuation  or  amount  is  usually  made  less  than  the  real 
value  of  the  property. 

The  insurance  business  is  chiefly  carried  on  by  corporations,  called 
Insurance  Companies.  In  Joint  Stock  Companies  the  profits  and  losses 
are  shared  by  the  stockholders,  but  in  Mutual  Companies  they  are 
divided  among  the  policy  holders. 

MENTAL    PKOBLEMS. 

1.  A  house  was  insured  for  $2500,  at  1  %  :   what  was 
the  premium? 

2.  A  stock  of  goods  was  insured  for  $8000,  at  f  %  :  what 
was  the  premium  ? 

3.  A  hotel  worth  $6000  is  insured  for  f  of  its  value,  at 
^i%''  what  is  the  premium? 

4.  What  will  it  cost  to  get  a  house  insured  for  $4000, 
for  10  years,  at  \%  a  year  ? 

5.  The  premium  paid  for  insuring  a  library  for  $500,  is 
$5 :  what  is  the  rate  of  insurance  ? 

6.  An  insurance  company  insures  a  school  house  for 
$10000,  and  charges  $50  premium:  what  is  the  rate? 

7.  The  premium  for  insuring  a  cargo  of  goods,  at  2%, 
was  $240  :  what  was  the  amount  of  goods  insured  ? 

-WRITTEN  PROBLEMS. 

8.  A  factory  worth  $75000  is  insured  for  f  of  its  value, 
at  1^  %  :  how  much  is  the  premium  ? 

9.  A  merchant  has  his  store  insured  for  $7850,  slI  ^%^^ 
and  his  goods  for  $12400,  at  |^  %  :  what  premium  does  he  pay  ? 


INSURANCE.  161 

10.  A  house  worth  $5400  was  insured  for  f  of  its  value, 
Sit  i%,  and  the  cost  of  the  survey  and  policy  was  $1.50: 
what  was  the  cost  of  the  insurance? 

Note. — When  a  new  risk  is  taken,  a  small  fee  is  usually  charged 
for  examining  the  property,  called  the  Survey,  and  for  issuing  the 
policy. 

11.  The  owners  of  a  vessel  paid  $561  for  a  hull  insurance 
of  $25500 :  what  was  the  rate  of  insurance  ? 

12.,  A  merchant  paid  $100  for  an  insurance  of  $12500 
on  a  stock  of  goods:  what  was  the  rate  of  insurance? 

13.  A  school  house  is  insured  at  |  %,  and  the  premium 
was  $93.60:   for  how  much  is  the  house  insured? 

14.  A  grain  shipper  paid  $525  for  the  insurance  of  a 
cargo  of  wheat,  at  1^  %  :  for  how  much  was  the  wheat  in- 
sured ? 

15.  A  company  insured  a  block  of  buildings  for  $150000, 
at  f  % ,  but,  the  risk  being  too  great,  it  re-insured  $40000  in 
another  company,  at  f  % ,  and  $35000  in  another  company, 
at  f  %.  How  much  premium  did  it  receive  more  than  it 
paid? 

16.  A  block  of  buildings  worth  $135000  is  insured  for  f 
of  its  value  by  three  companies,  the  first  taking  ^  of  the 
risk  at  f  %  ;  the  second  taking  f  of  it  at  f  %  ;  and  the 
third  taking  the  remainder  at  ^  %.  What  was  the  total 
premium  ? 

17.  Suppose  the  above  block  should  be  damaged  by  fire 
to  the  amount  of  $60000,  how  much  of  the  damage  would 
each  company  be  obliged  to  pay  ? 

18.  A  house  which  has  been  insured  for  $3500  for  10 
years,  at  f  ^  a  year,  was  destroyed  by  fire  :  how  much  did 
the  money  received  from  the  company  exceed  the  cost  of 
premiums  ? 

19.  A  steamer,  burned  in  1869,  had  been  insured  by  a 
single  company  20  years,  for  $40000,  at  24-  %  a  year :  what 
was  the  actual  loss  to  the  company,  no  allowance  being  made 
for  interest  ? 

20.  A  grain   dealer    had   a  cargo   of  wheat,    valued   at 

CAr.— 14. 


162  COMPLETE  ARITHMETIC. 

$31360,  insured  at  2  ^,  so  as  to  cover  both  the  value  of 
the  wheat  and  the  cost  of  the  premium :  for  how  much  was 
the  wheat  insured? 

Suggestion.— Since,  the  premium  was  2  %  of  the  amount  insured, 
the  vahie  of  the  wheat  was  100  % — 2  %,  or  98  %  of  the  amount  in- 
sured.    Hence,  $31360  =  x^ir  of  the  amount  insured.     (Case  IV.) 

21.  For  how  much  must  a  cargo  of  lumber,  worth  $21825, 
be  insured,  at  3  % ,  to  cover  both  the  value  of  the  lumber  and 
the  cost  of  the  premium  ? 

22.  For  how  much  must  property,  worth  $11859.40,  be 
insured,  at  1|-  % ,  to  cover  both  property  and  premium  ? 

23.  For  what  must  a  cargo  of  goods,  valued  at  $11520, 
be  insured,  at  4  % ,  to  cover  both  goods  and  premium  ? 

24.  What  amount  must  be  insured  to  cover  property 
worth  $2587,  and  premium  at  ^  %  ? 

25.  To  cover  both  goods  and  premium,  at  1%,  a  mer- 
chant had  a  cargo  of  goods  insured  for  $35000 :  what  was' 
the  value  of  the  goods  ? 

26.  A  merchant  shipped  a  cargo  of  flour  from  New  York 
to  Liverpool,  and,  to  cover  both  the  flour  and  the  premium, 
he  took  out  a  policy  for  $50400,  at  Z\%\  what  was  the 
value  of  the  flour  ? 

FORMULAS  AND  RULES. 

255.  Formulas.  — 1.  Premium  =  amount  insured  X  rate  % . 

2.  Bate  ^  :=prem.  -^  amount  insured. 

3.  Amount  insured  =prem.  -4-  rate  %. 

4.  Property  and  p)remium  =  propertij -^ 

(l  —  rate%). 

256.  Rules. — 1.  To  find  the  premium.  Multiply  the  amount 
insured  by  the  rate  per  cent,  expressed  decimally.     (Form.  1.) 

2.  To  find  for  what  sum  property  must  be  insured  to 
cover  both  property  and  premium.  Divide  the  value  of  the 
property  insured  by  1  less  the  rate  per  cent,  expressed  ded- 
mally.     (Form.  4.) 


LIFE  I^•SURANCE.  163 


LIFE  INSURANCE. 

257.  In  Life  Insurance  the  insurer  agrees  to  pay- 
to  the  heirs  of  the  insured,  or  to  some  person  named  in  the 
policy,  a  stipulated  sum  on  the  death  of  the  insured,  or  at 
a  specified  time,  should  his  death  not  occur  before. 

258.  When  the  policy  matures  at  the  death  of  the  in- 
sured, it  is  called  a  lAJe  Policy;  when  it  matures  in  a  speci- 
fied number  of  years,  it  is  called  a  Term  Policy  or  an  Endow- 
ment Policy, 

The  premium  in  life  insurance  may  be  paid  in  a  single  payment ; 
or  it  may  be  paid  annually  during  the  term  of  the  policy ;  or  it 
may  be  paid  annually  for  a  specified  number  of  years,  usually  for 
ten  years. 

A  Non-forfeiting  Policy  guarantees  the  insured  an  equitable  part  of 
the  sum  insured  in  case  he  should  fail  to  pay  his  annual  premiums 
after  a  specified  number  of  payments  have  been  made. 

259.  The  premium  is  computed  at  a  certain  sum  or  rate 
per  $1000  insured,  the  rate  varying  with  the  age  of  the  in- 
sured at  the  time  the  policy  is  issued. 

The  basis  on  which  the  rate  of  life  insurance  is  determined  is  the 
expectation  of  life,  or  the  average  extension  of  life  beyond  the  given 
age,  as  shoAvn  by  life  statistics.  Tables  have  been  formed  showing 
the  expectation  of  life  for  every  year  of  man's  age.     (See  appendix.) 

^^C^EITTEN  PEOBLEMS. 

27.  A  man  45  years  of  age  has  his  life  insured  for  $3000, 
at  $37.30  per  $1000:  what  annual  premium  does  he  pay? 

28.  A  man  30  years  of  age  has  his  life  insured  for  $6000, 
at  $23.60  per  $1000:  what  is  his  annual  premium? 

29.  A  man  38  years  of  age  is  insured  for  $5000  on  the  ten 
year  plan,  at  $44.50  per  $1000.  what  will  be  the  sum  of  his 
premiums  should  they  all  be  paid  ? 

30.  A  man  35  years  of  age  took  out  a  life  policy  for 
$4000,  at  the  rate  of  $27.50  per  $1000;  he  died  at  the  age 


164  •  COMPLETE  ARITHMETIC. 

of  60 :  how  much  greater  was  the  amount  insured  than  the 
sum  of  the  annual  paymfents  ? 

31.  A  man  27  years  of  age  took  out  a  life  policy  for 
$8000,  for  the  benefit  of  his  wife,  at  the  rate  of  $21.70 
per  $1000,  and  his  death  occurred  at  the  age  of  33:  how 
much  did  the  Avidow  receive  more  than  had  been  paid  in 
annual  premiums? 

TAXES. 

.260.  A  Tax  is  a  sum  of  money  assessed  on  the  person, 
property,  income,  or  business  of  a  citizen  for  the  support 
of  government  or  other  public  purposes. 

261.  A  tax  on  the  person  of  a  citizen  is  called  a  Poll 
TaXj  or  Capitation  Tax. 

262.  A  tax  on  property  is  called  a  Property  Tax.  It  is 
assessed  either  at  a  given  rate  per  cent,  of  the  valuation,  or 
at  the  rate  of  a  given  number  of  mills  on  the  dollar. 

Property  is  classified  as  Real  Estate  and  Personal  Property ^  the 
former  including  all  fixed  property,  as  houses  and  lands;  and  the 
latter,  all  movable  property.  The  taxable  value  of  real  estate  is 
appraised  by  oncers  called  Appraisers  or  Assesso7^s,  and  the  value 
of  personal  property  is  fixed  by  the  owner,  under  oath,  or  by  the 
assessor. 

263.  A  tax  on  the  annual  income  of  a  citizen  is  called 
an  Income  Tax ;  and  a  tax  on  his  business  is  called  an  Ex- 
cise Tax. 

Income  taxes  are  assessed  at  a  given  rate  per  cent,  of  a  citizen's 
net  income,  less  specified  exemptions  and  deductions.  Excise  taxes 
consist  of  fees  for  business  licenses,  revenue  stamps  for  business  papers, 
taxes  on  manufactured  products,  etc. 

The  Internal  Revenue  of  the  United  States  is  chiefly  derived  from 
income  and  excise  taxes,  assessed  and  collected  by  United  States  ofti^ 
cers,  called  Assessors  and  Collectors. 

Note. — Taxes  are  classified  as  direct  and  indirect.  Property  and 
poll  taxes  are  direct;  and  excise  taxes  and  duties  arc  indirect,  since 
they  are  paid  indirectly  by  the  consumer. 


TAXES.  165 


WBITTEN   PROBLEMS. 

1.  The  valuation  of  the  taxable  property  of  a  village  was 
$632000,  and  a  tax  of  $9480  was  assessed  to  build  a  school 
house :  what  was  the  rate  of  tax  ? 

Process.  Since  the  tax  was  .015  of  the  property, 

$9480 -H- $632000  r=  .015         the  rate  was  1.5^,  or   15  mills  on  the 
Kate  =  li%,  or  15  mills.      dollar. 

2.  The  tax  levied  in  a  certain  city,  for  all  purposes,  was 
$259776,  and  the  taxable  property  was  listed  at  $21648000 : 
what  w^as  the  rate  of  tax  in  mills? 

3.  The  amount  of  tax  to  be  assessed  in  a  certain  township 
is  $19340.20;  the  taxable  property  is  $1425400;  and  the 
number  of  polls,  assessed  at  $1.50  each,  is  540:  w^hat  rate 
of  tax  must  be  assessed  on  property? 

4.  The  cost  of  the  public  schools  of  a  certain  city  for  the 
next  school  year,  is  estimated  at  $36848 :  what  amount  of 
school  tax  must  be  assessed,  the  cost  of  collecting  being 
2^,  and  allowing  6  %  of  the  assessed  tax  to  be  uncol- 
lectible ? 

Process.  Since  2%  of  tax  collected  is  paid  for 

QQ  X  <|!q^o4o  collection,  $36848  is  98%  or  .98  of  the 

'm  )  $37600,  T<u  collected.      '"  '"  ^  collected.    S36848  is  .98  of 
$40000   ra.  a^,^.       «37600.     (Case  IV.)     Since  6  %  of  tl,e 
tax  assessed  is  not  collectible,  the  col- 
lectible tax,  or  $37600,  is  .94  of  the  tax  to  be  assessed.    $37600  is  .94 
of  $40000.     Hence,  $40000  is  the  amount  to  be  assessed. 

Note. — Since  the  amount  of  uncollectible  tax  can  only  be  esti- 
mated, the  amount  to  be  assessed  may  be  found,  with  sufficient  accu- 
racy for  all  practical  purposes,  by  adding  the  percentages  for  collec- 
tion and  for  uncollectible  taxes  to  the  amount  of  money  to  be  raised 
for  the  given  purpose. 

5.  The  net  proceeds  of  a  certain  tax  assessment,  after 
deducting  H%  for  collection,  was  $11703.84^,  and  H% 
of  the  tax  was  not  collected:  what  was  the  amount  of  tax 
assessed  ? 

6.  The  amount  of  tax  assessed  on  the  property  of  a  cer- 


166  COMPLETE   ARITHMETIC. 

tain  city  was  $145850;  the  treasurer  was  allowed  a  fee  of 
1%  for  collection,  and  10%  of  the  tax  was  uncollectible: 
what  were  the  net  proceeds  of  the  assessment? 

7.  The  taxable  property  of  a  certain  city  is  valued  at 
$87045060,  and  the  rate  of  tax  for  school  purposes  is  54- 
mills  on  the  dollar:  what  is  the  amount  of  school  tax  as- 
sessed ? 

vSuGGESTioN.— Since  5|  mills  =  .005|  of  a  dollar,  the  tax  assessed 
=  .005|  of  the  property. 

8.  The  valuation  of  taxable  property  in  a  certain  county 
is  $35460850,  and  the  rate  of  tax  levied  is  25  mills:  what 
will  be  the  net  proceeds  of  the  tax,  the  cost  of  collection 
being  3%,  and  8  %  of  the  tax  being  uncollectible? 

9.  When  the  rate  of  taxation  is  15  mills,  what  is  the 
amount  of  tax  on  A's  property,  listed  at  $13560?  On  B's, 
listed  at  $9850.60?     On  C's,  listed  at  $50060? 

10.  A  man's  net  income  is  $2750,  of  which  $1354  is  by 
law  exempt  from  taxation :  what  is  his  income  tax  at  5  %  ? 
At3%? 

11.  A  man's  income  is  $3570,  and  the  deductions  allowed 
by  law   amount  to  $1650:   what  is  his  income  tax  at  5  %  ? 

12.  A  man  pays  a  tax  of  Yl\  mills  on  his  property, 
listed  at  $9850,  and  an  income  tax  of  5  %  on  a  net  income 
of  $2750 :  what  is  his  total  annual  tax  ? 

FORMULAS  AND  RtJLES. 

264.  Formulas. — 1.  Tax  =^'pTOfe:riy  X  ^oie  %. 

2.  Uaie  %  =tax -i-  property. 

3.  Tax  collected  =  net  proceeds  -t-  (1  — 
rate  %  for  collection). 

4.  Tax  assessed  =  tax  collected  -f-  (1  — 
rate  %  of  tax  uncollected). 

265.  .Rules. — 1.  To  find  the  amount  of  tax,  Multiply  tJie 
amount  of  taxable  property  by  the  rate  of  tax,  expressed  ded- 
malhy. 


TAXES. 


167 


2.  To  find  the  rate  of  tax  in  mills,  Divide  the  amount  of 
tax  by  the  amount  of  property,  and  expi'ess  the  quotient  as  thou- 
sandths.    The  number  of  thousandths  will  be  the  number  of  mills. 

TAX  TABLES. 

266.  The  labor  of  making  out  a  tax  list  may  be  much 
lessened  by  using  tables  giving  the  tax  on  convenient 
amounts  of  property,  at  the  given  rate. 

Table  for  a  Bate  of  15  mills. 


PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

$1 

$.015 

$10 

$0.15 

$100 

$1.50 

$1000 

$15. 

$10000 

$150. 

2 

.03 

20 

.30 

200 

3.00 

2000 

30. 

20000 

300. 

3 

.045 

30 

.45 

300 

4.50 

3000 

45. 

30000 

450. 

4 

.06 

40 

.60 

400 

6.00 

4000 

60. 

40000 

600. 

5 

.075 

50 

.75 

500 

.   7.50 

5000 

75. 

50000 

750. 

6 

.09 

60 

.90 

600 

9.00 

6000 

90. 

60000 

900. 

7 

.105 

70 

1.05 

700 

10.50 

7000 

105. 

70000 

1050. 

8 

.12 

80 

1.20 

800 

12.00 

8000 

120. 

80000 

1200. 

9 

.135 

90 

1.35 

900 

13.50 

9000 

135. 

90000 

1350. 

13.  Find  by  the  above 
rate  of  15  mills. 


table  the  tax  on  $875.64,  at  the 


Process. 
$875.64=  $800  +  $70  +  $5  +  $.60  ■ 
$800.      --=$12.00 


70.   =   1.05 

Tax  on  J 

5.   =^   .075 

.60  =   .009 

[    .04  =   .0006 

Tax  on 

$875.64  =  $13.1346 

Since  $60  are  100  times 
60  cts.,  the  tax  on  60  cts. 
$.04.  is  found  by  dividing  the 
tax  on  $60  by  100,  which 
is  done  by  removing  the 
decimal  point  two  places 
to  the  left.  The  tax  on  4 
cts.  is  found,  in  like  man- 
ner, from  the  tax  on  $4. 


Find  by  the  above  table  the  tax  of 


14.  Mr.  A  on  $708.  19. 

15.  Mr.  B  on  $960.  20. 

16.  Mr.  C  on  $85.80.  21. 

17.  Mr.  D  on  $3405.  22. 

18.  Mr.  E  on  $860.50.  23. 


Mr.  F  on  $5408. 
Mr.  G  on  $85600. 
Mr.  H  on  $90908. 
Mr.  I    on  $150340. 
Mr.  J  on  $225350. 


168 


COMPLETE  ARITHMETIC. 


CUSTOMS  OR  DUTIES. 

267.  Customs  are  taxes  levied  by  the  national,  govern- 
ment on  imported  goods 
and  the  tonnage  of  ves- 
sels. Customs  are  also 
called   Duties. 

Points  of  Entry  for  foreign 
goods  are  established  by  law, 
and  at  each  port  of  entry  there 
is  a  Custom  House,  where  cus- 
toms or  duties  are  collected. 
The  officer  in  charge  of  the 
custom  house  is  called  the  Col- 
lector of  Customs,  and  a  list  of 
the  rates  of  duties  to  be  col- 
lected, is  called  a  Tariff. 

Duties  are  Specific  or  Ad 
Valorem. 

268.  Sjjecific  Duties  are  customs  assessed  on  the 
quantity  of  goods  imported,  without  reference  to  their 
value. 

In  assessing  specific  duties  an  allowance  is  made  (1)  for  waste,  called 
Draft;  (2)  for  the  weight  of  box,  cask,  etc.,  called  Tare  or  Tret;  (3)  for 
waste  of  liquids,  called  Leakage;  and  (4)  for  the  breaking  of  bottles, 
called  Breakage.  The  weight  of  goods  before  alloAvances  are  made  is 
called  Gross  Weight,  and  the  weight  after  all  allowances  are  made  is 
called  Net  Weight. 

269.  Ad  Valorem  Duties  are  customs  assessed  on 
the  cost  of  goods  in  the  country  from  which  they  are  im- 
ported. 

The  cost  of  imported  goods  is  sliown  by  an  Invoice  or  Manifest, 
and  when  the  currency  of  tlie  country  from  which  goods  are  im- 
ported has  a  depreciated  value,  the  amount  of  depreciation  is  stated 
in  a  consular  certificate,  attached  to  the  invoice.  When  the  owner 
or  consignee  can  not  exhibit  an  invoice  of  goods  at  the  custom  house, 
their  value  is  determined  by  appraisement. 


(CUSTOMS.  169 


^WRITTEN   PROBLEMS. 

1.  What  is  the  duty,  at  5  cts.  a  pound,  on  65  casks  of 
raisins,  gross  weight  115  lb.  each,  tare  12%? 

2.  What  is  the  duty,  at  25  cts.  a  pound,  on  1240  chests 
of  tea,  gross  weight  120  lb.  each,  tare  10  %  ? 

3.  What  is  the  duty,  at  5  cts.  a  pound,  on  340  sacks  of 
coffee,  250  lb.  ^ross  each,  tare  5  %  ? 

4.  What  is  the  duty,  at  14-  cts.  a  pound,  on  240  tons  of 
bar  iron,  draft  5  %  ? 

Note. — In  custom  house  computations  a  cwt.  =  112  lb. 

5.  A  merchant  imported  from  Havana  225  hogsheads  of 
sugar,  475  gross  each,  tare  12|-  %  ;  and  120  hogsheads  of 
molasses,  126  gal.  each,  leakage  2  %  '  what  was  the  duty, 
at  3  cts.  a  lb.  for  sugar,  and  8  cts.  a  gal.  for  molasses? 

6.  A  merchant  imported  a  lot  of  silks,  invoiced  at  $45360: 
what  was  the  duty,  at  50  %  ad  valorem  ? 

7.  A  merchant  imported  1450  yards  of  broadcloth,  in- 
voiced at  $2.15  a  yd.  ;  3240  yards  Brussels  carpeting,  in- 
voiced at  $1.60  a  yd. ;  and  480  yards  of  silk,  invoiced  at 
$2.85:  how  much  was  the  duty,  at  35%  for  the  woolen 
goods,  and  50  %  for  the  silk  ? 

8.  The  duty  on  1250  yards  of  silk,  at  40%  ad  valorem, 
was  $1100:  what  was  the  invoice  price  a  yard?  For  how 
much  a  yard  must  the  importer  sell  the  silk  to  clear  20  %  ? 

FORMULAS  AND  RULES. 

270.  Formulas. — 1.   Specific  duty  =  net  quantity  y^  rate  %. 

2.  Ad  val.  duty = net  inv.  price  X  t'cite  % . 

3.  Net  mv.  price  ==  ad  val.  duty  h-  rate  % . 

271.  KuLES. — 1.  To  find  specific  duty,  Multiply  the  num- 
ber, denoting  the  net  quantity  of  the  goods,  by  the  rate  per  cent. , 
or  by  the  duty  on  one. 

2.  To  find  ad  valorem  duties,  Multiply  the  invoice  price  less 
deductions  allowed,  by  the  rate  per  cent,  expressed  decimally. 

C.Ar.— 15. 


170  COMPLETE    ARLTHMKTIC. 


BANKRUPTCY. 

272.  A  Bankrupt  is  a  person  who  fail::  in  business 
and  has  not  property  enough  to  pay  all  his  debts.  A  ])ank- 
rupt  is  also  called  an  Insolvent. 

Note. — The  lerui  ])ankrupt  is  strictly  applicable  only  to  a  trader, 
wiiile  the  term  insolvent  applies  to  any  ]ierson  wlio  is  unable  to  pay 
his  debts. 

273.  Bankvaptcy  is  a  failure  in  business,  with  ina- 
bility to  pay  all  debts. 

274.  An  Assif/nnient  is  the  transfer  of  the  property 
of  a  bankrupt  to  certain  persons,  called  Assignees,  in  whom  it 
is  vested  for  the  benefit  of  the  creditors. 

Note. — It  is  the  duty  of  assignees  to  convert  the  property  into 
money  and  divide  the  proceeds,  alter  deducting  expenses,  among  the 
creditors. 

275.  The  property  of  a  bankrupt  or  insolvent  is  called 
his  Assets,  and  the  amount  of  his  indebtedness  is  called  his 
Liabilities.  The  assets  less  the  expense  of  settling  are  the 
Net  Proceeds. 

WRITTEN  PKOBLEMS. 

1.  A  merchant  failed  in  business,  owing  $15750,  and  his 
assets  amount  to  $10515:  what  per  cent,  of  his  liabilities 
can  he  pay,  allowing  $750  for  expense  of  settling. 

Process.  Since  the  net  proceeds  of 

$10515  —  $750  =  $9765,  7iet  proceeds.        ^'^  ^^•''ets  are  but  .62  of  his 

$9765 -- $15750  =  .62,  or  62  %  liabilities,    he   can    pay    but 

C)2fc,  or  62  cts.  on  a  dollar. 

2.  In  the  above  case  of  bankruptcy  there  are  four  cred- 
itors, whose  claims  are  respectively,  $3580,  $4635,  $5300, 
and  $2235 :  how  much  will  each  receive  ? 

3.  Smith,  Jones  &  Co.  have  become  insolvent,  owing  A 
$3500,  B  $1250,  C  $3750,  D  $1000,  and  E  $2500;  their 
assets  amount  to  $7150,  and  the  expense  of  settling  w^ill 
be  $550:  what  per  cent,  of  their  liabilities  can  they  pay? 
What  will  each  creditor  receive? 


INTEREST.  171 

4.  A  dry  goods  merchant  failed,  with  liabilities  amount- 
ing to  $25000;  his  assets  are:  goods  $9500,  building  and 
lot  $5400,  and  bills  collectible  $2100 ;  and  the  expense  of 
settling  will  be  5  %  of  the  assets.  How  many  cents  on  the 
dollar  can  he  pay? 

FORMULAS  AND  RULES. 

276.  Formulas. — 1.  Rate  %  =  net  'proceeds  -i-  liabilities. 

2.   Dividend  =  claim  X  "i'ate  %. 

277.  Rules. — 1.  To  find  what  per  cent,  of  his  liabilities 
a  bankrupt  can  pay,  Divide  the  net  proceeds  of  his  assets  by 
the  amount  of  his  liabilities,  and  the  quotient  expressed  in  hun- 
dredths  ivHl  be  the  rate  per  cent. 

2.  To  find  the  dividends  of  creditors  in  a  case  of  bank- 
ruptcy, Multiply  the  several  claims  of  creditors  by  the  rate  per 
cent,  which  the  net  jwoceeds  of  the  assets  ivill  pay. 

Note. — It  is  more  common  to  find  how  many  cents  on  the  dollar 
the  net  proceeds  will  pay;  the  process  is  the  same. 

INTEREST. 

PRELIMINARY  DEFINITIONS. 

278.  Interest  is  the  premium  paid  for  the  use  of  money. 

279.  The  Principal  is  the  sum  of  money  for  the  use 
of  which  interest  is  paid. 

280.  The  Amount  is  the  sum  of  the  principal  and  in- 
terest. 

281.  The  Hate  of  Interest  is  the  number  of  hun- 
dredths of  the  principal  paid  for  its  use  one  year. 

282.  The  rate  of  interest  fixed  by  law  is  called  the  legal 
rate;  and  any  rate  of  interest  higher  than  the  legal  rate  is 
usury. 

The  legal  rate  of  interest  in  most  of  the  states,  and  on  debts  due 
the  United  States,  is  6%.  In  several  states  a  rate  higher  than  the 
legal  rate  is  allowed,  when  so  stipulated  in  the  contract.     (Art.  438.) 


172  COMPLETE  ARITHMETIC. 

283.  Simple  Interest  is  interest  on  the  principal  only. 

Interest  considers  the  element  of  time,  in  which  respect  it  differs 
from  the  previous  applications  of  percentage.  For  periods  of  time 
greater  or  less  than  one  year,  the  interest  is  proportionally  greater  or 
less  than  the  interest  for  one  year. 


GENERAL  METHOD  OF  COMPUTING  INTEREST. 

MENTAL   PROBLEMS. 

1.  When  money  is  loaned  at  6  %  a  year,  what  part  of 
the  principal  equals  the  interest  for  one  year? 

Ans. — Since  6  %  =:  .06,  the  interest  for  one  year  equals  .06  of  the 
principal. 

2.  What  part  of  the  principal  equals  the  interest,  when 
money  is  loaned  at  5  %  ?     At  8  %  ?     At  10  %  ? 

3.  What  part  of  the  principal  equals  the  interest,  when 
money  is  loaned  at  4  %  ?     At  5|-  %  ?     At  7^  %  ? 

4.  What   is  the   interest  of  $50  for  one  year  at  6  %  ? 

Solution. — Since  the  interest  for  one  year,  at  6  %,  equals  .06  of 
the  principal,  the  interest  of  $50  for  one  year  equals  .06  of  $50,  which 
is  $3. 

5.  What  is  the  interest  of  $400  for  one  year  at  7  %  ? 
At  8  %  ?    At  9  %  ?    At  10  %  ? 

6.  What  is  the  interest  of  $650  for  one  year,  at  4  %  ? 
At  6  %  ?     At  8  %  ? 

7.  What  is  the  interest  of  $120  for  one  year,  at  5  %  ? 
At  51  %  ?     At  10  %  ? 

8.  What  is  the  interest  of  $250  for  3  years,  at  6  %  ? 

Solution.— The  interest  of  $250  for  1  year,  at  6%,  is  $15,  and 
since  the  interest  for  1  year  is  $15,  the  interest  for  3  years  is  3  times 
$15,  which  is  $45. 

9.  At  1%,  what  is  the  interest  of  $300  for  4  years? 
For  5  years?     10  years? 

10.  At  8%,  what  is  the  interest  of  $150  for  2  years? 
4^  years  ?     5^  years  ?     8  years  ? 


INTEREST.  173 

11.  At  44%,  what  is  the  interest  of  $200  for  3  years? 
4i  years  ?     Q^  years  ? 

12.  At  10  %,  what  is  the  interest  of  $25  for  6  years?     12 
years  ?     8f  years  ? 

13.  What  is  the  interest  of  $70  for  2  years  and  4  months, 
at  5  %  ? 

Suggestion. — 4  months  =  i  of  a  year,  and  2  yr.  4  mo.  =  2^  yrs. 

14.  At  4%,  what   is   the   interest   of  $15   for  3  years  3 
months?     For  5  years  6  months? 

15.  At  7  %,  what  is  the  interest  of  $30  for  2  years  4 
months?     3  years  2  months? 

16.  At  6  %,  w^hat  is  the  interest  of  $50  for  4  years  2 
months?     6  years  10  months? 

17.  What  is  the  interest  of  $10  for  4  years  6  months,  at 
4  %  ?     At  6  %  ?     At  9  %  ?     At  10  %  ? 

18.  What  is  the  interest  of  $500  for  3  years  2  months,  at 
5%?    At8%?    At  12%? 

WRITTEN  PROBLEMS. 

19.  What  is  the  interest  of  $145.60  for  5  years  10  months, 
•at  5  %  ? 

Process. 

$145.60 
.05 


$7.2800= /we. /or  1  year. 
54 


36.40 
6.07 


$42.47  =  Int.  for  5  yr.  10  mo. 

20.  What  is  the  interest  of  $273.45  for  8  years  3  months, 
at  7i  %  ?     At  10  %  ? 

What  is  the  interest  of 

21.  $65.30  for  1  yr.  3  mo.,  at  6  %  ?     At  8  %  ? 

22.  $640.58  for  4  yr.  11  mo.,  at  5  %  ?     At  10  %  ? 

23.  $1000  for  1  yr.  1  mo.,  at  3|  %  ?     At  7.3  %  ? 

24.  $85,  at  7  %,  for  3  yr.  7  mo.  ?     For  9  months? 


174  COMPLETE  ARITHMETIC. 

25.  $38.10,  at  9  %,  for  6  yr.  5  mo.  ?     For  3  yr.  10  mo.? 

26.  $84.75  for  2  yr.  5  mo.  21  da.,  at  8  %  ? 

Suggestion. — Reduce  the  5  mo.  21  da.  to  the  decimal  of  a  year. 
(Art.  171,  N.  2.)  21  da.  =  .7  mo.,  and  5.7  mo.  =  .475  yr.  Hence, 
2  yr.  5  mo.  21  da.  =2.475  yr. 

27.  $208.44  for  7  yr.  8  mo.  15  da.,  at  5  %  ? 

28.  $356.75  for  5  yr.  10  mo.  24  da.,  at  6^  %  ? 

29.  $184.80  for  1  yr.  1  mo.  10  da.  (1^  yr.),  at  9  %  ? 

30.  $321.70  for  4  yr.  3mo.  27  da.,  at  12|-  %  ? 

What  is  the  amount  of 

31.  $60.85  for  10  yr.  10  mo.  10  da.,  at  10  %  ? 

32.  $740.10  for  1  yr.  1  mo.  18  da.,  at  8  %  ? 

33.  $1.40  for  7  yr.  11  mo.  21  da.,  at  7|  %  ? 


34.  $121.75  for  3  yr.  18  da.,  at  12 


35.  $80.65  for  1  yr.  6  mo.  12  da.,  at  10^  %  ? 

36.  What  is  the  interest  of  $356.50  for  3  yr.  9  mo.  25 
da.,  at  8  %? 


Process  by  Aliquot  Parts 

FOR  Days. 

$356.50 
.08 

12  )  $28.5200  X  3  :: 

=  $85,560 

Int. 

for 

3  yrs. 

to.)     $2.3766X9  = 

=   21.389 

<( 

u 

9?no. 

r^mo. 

1.188 

a 

li 

15  da. 

r  1  mo. 

.792 

li 

u 

10  da. 

{Int  for  1 
15  da. : 
10  da. : 

$108,929     Int.  for  3  yr.  9  mo.  25  da. 

What  is  the  interest  of 

37.  $84.66  for  5  yr.  7  mo.  20  da.,  at  5  %? 

38.  $4000  for  10  yr.  10  mo.  10  da.,  at  15%? 

39.  $1262.70  for  11  mo.  27  da.,  at  7^  %  ? 

40.  $504.08  for  3  yr.  1  mo.  1  da.,  at  10  %  ? 

41.  $3084.90  for  7  mo.  22 .da.,  at  12  %  ? 

42.  $2016.05  for  1  yr.  1  mo.  29  da.,  at  8  %  ? 

43.  What  is  the  amount  of  $262.75  for  1  yr.  5  mo.  19 
da.,  at  6%?    At7%?     At  9  %  ? 

44.  What  is  the  amount  of  $192.60  for   2  yr.  2  mo.  2 
da.,  at  5%?     At  10%?     At  12%? 


INTEREST.  17& 

45.  A  man  borrowed  $60  May  10,  1864,  and  paid  it 
March  4,  1866,  with  interest  at  6  %  :  what  amount  did  he 
pay? 

Suggestion. — Find  the  difference  of  time  by  compound  subtrac- 
tion. 

46.  What  is  the  interest  of  $15.80,  from  Oct.  23,  1855, 
to  Apr.  12,  1859,  at  8  %  ? 

47.  A  note  of  $565.80,  dated  June  3,  1864,  was  paid  Nov. 
28,  1869,  with  interest  at  8  %  :  what  was  the  amount  paid? 

PRINCIPLE,  FORMULA,  AND  RULE. 

284.  Principle.  —  The  principal  multiplied  by  the  rate  per 
cent,  equals  the  interest  for  one  year. 

285.  Formula. — Interest  =  principal  X  rate  %  X  time. 

286.  Rule. — To  find  the  interest  of  any  sum  of  money 
for  any  time,  at  any  rate  per  cent.,  1.  Multiply  the  principal 
by  the  rate  per  cent. ,  expressed  decirhally,  and  multiply  this  pro- 
duct by  the  time  in  years  and  the  fraction  of  a  year.     Or, 

2.  Multiply  the  interest  for  one  year  by  the  number  of  years, 
and  -^  of  it  by  the  number  of  months,  and  find  the  interest  for 
days  by  aliquot  parts.  The  sum  of  the  several  results  will  be  the 
interest  for  the  given  time. 

Note. — In  solving  the  majority  of  problems  in  interest,  the  re- 
duction of  the  months  and  days  to  the  decimal  of  a  year  will  be 
found  as  brief  as  the  method  by  aliquot  parts.  Those  who  prefer  to 
use  aliquot  parts,  will  find  the  method  given  above  briefer  than  the 
one  generally  used. 


SIX  PER  CENT.  METHOD. 

MENTAL   PROBLEMS. 

1.  What  is  the  interest  of  $1  for  2  years,  at  6  %  ? 

Solution.— The  interest  of  $1  for  1  year  at  6%  is  .06  of  $1, 
which  is  6  cents,  and  the  interest  for  2  years  is  2  times  6  cents,  which 
is  12  cents. 


176  COMPLETE  ARITHMETIC. 

2.  What  is  the  interest  of  $1,  at  6  %,  for  3  years?     For 
8  years?     12  years?     15  years? 

3.  What  is  the  interest  of  $1,  at  6  %,  for  5  years?     For 
10|^  years?     16 1  years? 

4.  What  is  the  interest  of  $1  for  1  month,  at  6  %  ?    For 
3  months? 

Solution. — Since  the  interest  of  $1  for  1  year,  at  6^,  is  6  cents, 
the  interest  for  1  month  is  -^ 2^  of  6  cents,  which  is  5  mills,  and  the 
interest  for  3  months  is  3  times  5  mills,  which  is  15  mills. 

5.  AVhat  is  the  interest  of  $1,  at  6  %,  for  4  months?     6 
months?     8  months?     10  months? 

6.  What  is  the  interest  of  $1,  at  6  %,  for  5  months?     7 
months?     9  months?     11  months? 

At  6  %,  what  is  the  interest  of 

7.  $1  for  1  year  2  months  ?     2  yr.  4  mo.  ?     3  yr.  6  mo.  ? 

8.  $1  for  4  years  5  months  ?     6  yr.  7  mo.  ?    5  yr.  9  mo.  ? 

9.  $1  for  2  yr.  11  mo.  ?     3  yr.  9  mo.  ?     10  yr.  10  mo.  ? 

10.  What  is  the  interest  of  $1,  at  6  %,  for  1  day?  For 
6  days? 

Solution. — Since  the  interest  of  $1  for  80  days,  at  6%,  is  5  mills, 
the  interest  for  1  day,  is  y^  of  5  mills,  which  is  -g-  of  1  mill,  and  the 
interest  for  6  days  is  6  times  ^  of  1  mill,  which  is  1  mill. 

11.  What  is  the  interest  of  $1,  at  6  %,  for  12  days?  For 
18  days?     For  24  days? 

12.  What  is  the  interest  of  $1,  at  6%,  for  9  days?  15 
days?     21  days?     27  days? 

13.  What  is  the  interest  of  $1,  at  6  %,  for  10  days?  20 
days?     14  days?     22  days? 

14.  What  is  the  interest  of  $1,  at  Q%,  for  7  days?  11 
days  ?     17  days  ?     23  days  ?     25  days  ? 

15.  What  is  the  interest  of  $1,  at  6%,  for  2  months  12 
days?     4  mo.  18  da.?     5  mo.  6  da.? 

At  6  % ,  what  is  the  interest  of 

16.  $1  for  6  mo.  9  da?     8  mo.  15  da.?     10  mo.  21  da.? 

17.  SI  for  5  mo.  13  da.  ?     3  mo.  22  da.  ?     7  mo.  25  da.  ? 


INTEREST.  177 

18.  $1  for  4  mo.  16  da.  ?     6  mo.  29  da.  ?     10  mo.  10  da.  ? 

19.  $1  for  6  mo.  16  da.  ?     9  mo.  28  da.?     11  mo.  11  da.? 

20.  $1  for  2  yr.  8  mO.  12  da.  ?     4  yr.  5  mo.  18  da.  ? 

21.  $1  for  3  yr.  3  mo.  24  da.  ?    5  yr.  6  mo.  6  da.  ? 

22.  At  6^,  what  is  the  interest  of  $1  for  1  year?     For 
1  month  ?     2  months  ?     For  1  day  ?     6  days  ? 

23.  If  the  interest  of  a  certain  principal  is  $12,  at  6  %, 
what  would  be  the  interest  of  it  at  7  %  ?     At  8  %  ? 

Suggestion.— 7^  is  ^  more  than  6%;  and  8%  is  |  more  than  6%. 

If  the  interest  of  a  certain  principal,  at  6  %,  is  $36,  what 
would  be  the  interest  of  it 


24. 

At  12  %  ? 

At  15  %  ? 

At  18%? 

At  21%? 

25. 

At    3%? 

At    4%? 

At4i%? 

At    5%? 

26. 

At  10^? 

At  11%? 

At  13%? 

At  14%? 

WBITTEN   PROBLEMS. 

27.  What   is   the   interest   of  $245.60   for   2   yr.  7  mo. 
21  da.,  at  6%? 

Process. 

Int.  of  $l  =  $.158i  Since  the   interest   of  %\   for   2  yr.  7  mo. 

$245.60  21  da.,  at  6%,  is  $.158i,  or  .158^  of  $1,  the  in- 

.1581  terest  of  $245.60  will  be  .158|  of  $245.60,  which 

196480  is  $38,928.     The  interest  is  as  many  thousandths 

i??^2*^  ^^  *he  principal,  as  the  interest  of  $1  is  thou- 
sandths of  $1. 


24560 
12280 
$38.9276,  Int. 


28.   What  is   the  interest   of  $245.60   for   2   yr.   7  mo. 
21  da.,  at  9%?     At  11  %  ? 

Process.  Process. 

$38,928,  Int.  at  6%.  $38,928,  Int.  at    6%. 

19.464,     "     "  3%.  19.464,     "     "     3%. 

$58,392,     "     ''9%.  12.976,     "     "     2%. 

$71,368,     "     "  11%. 


178  COMPLETE  ARITHMETIC. 

29.  What  is  the  interest  of  $508.09  for  3  yr.  3  mo. 
15  da.,  at  6%?     At  5  %  ?     At  7  %  ? 

What  is  the  interest  of 

30.  $540  for  10  mo.  24  da.,  at  6  %?     At  8  %? 

31.  $327.50  for  1  yr.  3  mo.  6  da.,  at  7  %?    At  10%? 

32.  $142.64  for  2  yr.  15  da.,  at  4  %  ?     At  4i  %  ? 

33.  $3008.75  for  4  yr.  1  mo.  20  da.,  at  5  %  ?     At  9  %  ? 

34.  $622,40  for  9  mo.  29  da.,  at  12  %  ?     At  15  %  ? 

What  is  the  amount  of 

35.  $804.25  for  1  yr.  5  mo.  10  da.,  at  8  %  ?     At  74-  %  ? 

36.  $112.40  for  11  mo.  21  da.,  at  5^%?     At  H%? 

37.  $2000  for  1  yr.  1  mo.  1  da.,  at  8  %  ?     At  11  %  ? 

38.  $5.90  for  3  yr.  3  mo.  3  da.,  at  3  %  ?     At  12  %  ? 

39.  $16.50  for  2  yr.  2  mo.  2  da.,  at  6  %  ?     At  7|  %  ? 

40.  $50.30  for  3  yr.  3  mo.  3  da.,  at  8  %  ?     At  5  %  ? 

41.  $200  for  4  yr.  4  mo.  4  da.,  at  4  %  ?     At  10  %  ? 

42.  What  is  the  interest  of  $108.60,  from  Sept.  12,  1866, 
to  May  6,  1870,  at  6  %  ?     At  8  %  ? 

43.  A  debt  of  $40.50  was  paid  May  21,  1870,  with  in- 
terest, at  6%,  from  Nov.  9,  1864:  what  was  the  amount 
paid? 

44.  A  note  of  $350,  dated  Oct.  17,  1865,  was  paid  Apr. 
11,  1868,  with  interest  at  7  % :  what  was  the  amount  paid? 

45.  A  note  of  $150.75,  dated  June  15,  1867,  was  paid 
Jan.  1,  1870,  with  interest  at  5  %  :  what  was  the  amount 
paid  ? 

46.  A  note  of  $1250,  dated  July  5,  1868,  was  paid 
June  1,  1870,  with  interest  at  8  % :  what  was  the  amount 
paid? 

47.  A  note  of  $87.50,  dated  Aug.  8,  1867,  and  bearing 
interest  at  10  %,  was  paid  March  25,  1869  :  what  was  the 
amount  paid? 

48.  A  note  of  $65.80,  dated  Feb.  20,  1868,  and  bearing 
interest  at  7%,  was  paid  June  25,  1870:  what  was  the 
amount  paid? 


INTEREST.  179 

FORMULA  AND  RULES. 

287.  Formula. — Int.  at  6%=  principal  X  int.  of  $1  atQ%. 

288.  Rules. — 1.  To  compute  interest  at  6%,  Find  the 
interest  of  $1  for  the  given  time,  by  takiiig  six  times  as  Tnany 
cents  as  there  are  years,  one-half  as  many  cents  as  there  are 
moniJis,  and  one-sixth  as  many  mills  as  there  are  days;  and 
then  midtiply  the  principal  by  the  abstract  decimal  which  corre- 
sponds to  the  interest  of  %\  tlius  found. 

'  2.  To  compute  interest  at  any  other  rate  than  6  % ,  Find 
the  interest  at  6  %,  and  then  increase  or  diminish  tJiis  interest  by 
sudi  a  part  of  itself  as  will  give  Hie  interest  at  Hie  given  rate. 

METHOD  BY  DAYS. 

289.  When  the  time  is  short,  it  is  the  custom  of  bankers 
and  other  business  men  to  compute  interest  for  the  actual 
number  of  days  included  in  the  time,  each  day  being  con- 
sidered as  -^  of  a  year. 

'WKITTEN   PROBLEMS. 

1.  What  is  the  interest  of  $80.60  from  March  15th  to 


Allowing  360  days  to  a  year, 
the  interest  for  87  days  is  -^-^-^  of 
the  interest  for  one  year,  and  the 
interest  for  1  year,  at  6%,  is  jIj^ 
of  the  principal.  Hence,  the  in- 
terest for  87  days  is  /g^  of  j^-^ 
of  the  principal,  which  is  (j§^^ 
of  the  principal.  But  ^§^^  =  ^ 
of  T§^7-  Hence,  the  interest,  at  6  %,  equals  one-sixth  as  many  thou- 
sandths of  the  principal  as  there  are  days  in  the  time. 

Note. — This  explanation  may  be  preferred :  Allowing  360  days  to 
the  year,  the  interest  of  $1  for  1  day,  at  6%,  is  j\^  of  6  cents,  or  60 
mills,  which  is  I  of  a  mill,  and,  hence,  the  interest  of  $1  for  any  num- 
ber of  days  is  |  of  as  many  mills  as  there  are  days.  Having  found  the 
interest  of  $1,  the  interest  of  any  principal  is  found  as  in  the  preced- 
ing article. 


June  10th,  at  6%? 

Process  : 

In  March  16  days. 

$80.60 

«    Apr.     30     " 

.014^ 

''    May     31      " 

32240 

"    June     10     " 

8060 

6  )  87  days. 

4030 

m 

$1.16870 

180  COMPLETE    ARlTHMEi'IC. 


2.  What  is   the  interest  of   $125.80  from  July  5th  to 
,.  23d,  at  6  %  ?     At  8  %  ? 

3.  What  is  the  amount  of  $25.25  from  Oct.   30,  1869, 

9 


to  Feb.  1,  1870,  at  6  %?     At  7^% 

4.  What  is  the  amount  of  $65.80  from  Dec.  28,  1867, 
to  Mch.  15,  1868,  at  5  %  ?    At  10  %  ? 

5.  What  is  the  amount  of  $75.40  from  Jan.   13,  1869, 
to  June  15,  1869,  at  6  %  ?     At  7  %  ? 

6.  What  is  the  amount  of  $120  from  Mch.  15,  1870,  to 
July  4,  1870,  at  7  %  ?     At  9  %  ? 

7.  A  note  of  $420,  dated  Jan.  25,  1860,  was  paid  Apr. 
16,  1860,  with  interest  at  8  %  :   what  was  the  amount? 

8.  A  man  borrowed  $150,  June  6th,  and  paid  it,  with 
interest  at  7  %,  Sept.  24th:   how  much  did  he  pay? 

9.  A  note  of  $80,  dated  Jan.  15,  1868,  was  paid  June 
21,  1868,  with  interest  at  8  %  :   what  was  the  amount? 

10.  A  note  of  $150,  dated  Mch.  30,  1870,  was  paid  July 
4,  1870,  with  interest  at  6  %  :  what  was  the  amount? 

11.  A  note  of  $500,  dated  May  12,  1869,  and  bearing 
interest  at  7  %,  was  paid  July  24,  1869:  what  was  the 
amount? 

.     FORMULA  AND  RULES. 

290.  Formula. — Interest  at  6%  ^=^  principal  X  daijs  -f-  6000. 

291.  Rules. — 1.  To  compute  interest  for  days  at  6  %, 
Multiply  the  principal  by  one  sixth  of  as  many  thousandtlis  as 
there  are  days  in  the  time. 

2.  To  compute  interest  for  days  at  any  %,  Find  the  in- 
terest at  Q  ^,  and  then  increase  or  diminish  this  hiterest  by  such 
a  part  of  itself  as  the  given  rate  is  greater  or  less  than  6. 

Notes. — 1.  Since  the  common  year  consists  of  365  days,  instead 
of  360,  the  true  interest  for  360  clays  is  fff  or  f|  of  the  interest  for 
a  year ;  whereas,  by  the  above  method,  the  interest  for  360  days  equals 
the  interest  for  a  year.  Hence,  tlie  true  interest  for  any  number  of 
days  in  a  common  year  is  i^\  less  than  the  interest  found  by  the  above 
rule ;  in  leap  year  the  true  interest  is  ^'y  less  than  the  interest  thus 
found.     An  accurate  rule  for  computing   interest  for  days  is,  to  take 


PARTIAL  PAYMENTS.  181 

OS  many  36btks,  and  in  leap  year  as  many  S66ths,  of  the  interest  for  one 
year  as  there  are  days  in  the  time. 

2.  In  Great  Britain,  a  day's  interest  is  made  ^^-^  of  a  year's  inter- 
est, and  the  same  rule  is  adopted  by  the  United  States  Government  in 
computing  interest  upon  bonds,  etc.  The  convenience  of  the  method 
whicli  allows  360  days  to  the  year  has  secured  its  very  general  adop- 
tion by  the  business  men  of  the  country,  and  in  several  states  it  is 
sanctioned  by  law. 

3.  There  are  three  methods  of  finding  the  time  between  two  dates, 
to- wit:  1.  By  compound  subtraction,  allowiny  30  days  to  the  month.  2. 
By  finding  the  number  of  calendar  months  from  the  first,  date  to  the  corre- 
sponding day  of  the  month  of  the  second  date^  and  then  counting  the  actual 
number  of  days  left.  3.  By  counting  the  actual  number  of  days  between 
the  two  dates.  The  third  metliod  is  strictly  accurate,  and  is  generally 
used  in  finding  the  time  of  "  short  paper."  The  number  of  days  may 
be  found  from  *'  Time  Tables,"  which  give  the  exact  number  of  days 
between  any  two  dates  less  than  a  year  apart. 


PARTIAL  PAYMENTS. 

292.  When  partial  payments  have  been  made  on  notes  and 
other  obligations,  the  interest  is  computed  by  the  following 
rule,  which,  having  been  adopted  by  the  Supreme  Court  of 
the  United  States,  is  called  the 

UNITED  STATES  RULE. 

When  partial  payments  have  been  made,  app)ly  the  payment, 
in  the  first  place,  to  the  discharge  of  the  interest  then  due.  If 
the  payment  exceeds  the  interest,  the  surplus  goes  toward  dis- 
charging the  principal,  and  the  subsequent  interest  is  to  be  coin- 
puted  on  the  balance  of  principal  remaining  due. 

If  the  payment  be  less  than  the  interest,  the  surplus  of  inter- 
est must  not  be  taken  to  augment  the  principal,  but  interest  con- 
tinues on  the  former  principal  until  the  period  ivhen  the  2:)ay- 
ments,  taken  togetJier,  exceed  tlw  interest  due,  and  tJien  the  sur- 
plus u  to  be  applied  toward  discharging  the  principal,  and  the 
interest  is  to  be  computed  on  the  balance,  as  aforesaid. 

293.  This  rule  requires,  first,  that  payments  be  applied 
to  the  discharge  of  interest  then  due ;  and,  secondly,  that 
no  unpaid  interest  be  added  to  the  principal  to  draw  inter- 
est.    Interest  accrues  only  on  the  unpaid  pinncipal. 


182  COMPLETE  ARITHMETIC. 


PROBLEMS. 

1.  A  note  of  $650,  dated  May  20,  1866,  and  drawing  in- 
terest at  6  % ,  had  payments  indorsed  upon  it  as  follows : 

Sept.  2,  1866,  $2d.  March  2,  1867,  $150. 

Dec.  20,  1866,  $10.  July  8,  1867,  $200. 

What  was  the  amount  due  Nov.  11,  1867? 

Process. 

$650 
.017 
4550 
1866        9  2  650 

1866         5 20  $11,050      Ist  interest. 

.017  650. 

661.05 


3  mo.  12  da. 

$25. 

1866 
1866 

12    20 

9    2 

3  mo.  18  da. 

$10. 

1867 
1866 

3    2 
12    20 

2  mo.  12  da. 

$150. 

1867 
1867 

7    8 
3    2 

4  mo.  6  da. 

$200. 

1867 
1867 

11    11 

7    8 

25.00      1st  payment. 
$636.05     2d  principal. 
.018 


.018  508840 

_63605_ 
$10     $11.44890      2d  interest. 


$636.05     Zd  principal. 
.012 
.012  $150      $7^63260      M  interest. 

11.4489        2d  interest. 
636.05 
$655.1315 

160.00  2d  +  Sd  payment. 

$495.1315       4th  principal. 
.021  .021 


4951315 
9902630 


$10.3977615       4.th  interest. 

495.1315 

4  mo.    3  da.    .0205  $505.5293 

200.00  4th  payment. 

$305.5293  bth  principal. 

.0205 
15276465 
6110586 


$6.26335065       bth  interest. 
305.5293 


$311.7926,  Amount  dne  Nov.  11,  1867. 


PARTIAL  PAYMENTS.  183 

The  first  step  is  to  find  the  difierence  of  time  between  each  two 
consecutive  dates,  and  form  the  corresponding  decimal  multipliers 
by  the  six  per  cent,  method  (Art.  288).  The  payments  may  be 
written  below.  This  preparation  will  lessen  the  liability  of  error  in 
the  calculation. 

Since  the  1st  payment  is  greater  than  the  1st  interest,  form  the 
amount  and  subtract  therefrom  the  payment.  The  difference  is  the 
2d  principal.     Find  the  2d  interest. 

Since  the  2d  payment  is  less  than  the  2d  interest,  let  the  interest 
stand,  drawing  a  double  line  beneath  it,  and  bringing  down  the  2d 
principal  for  a  3d  principal.     Find  the  3d  interest. 

Since  the  sum  of  the  2d  and  3d  payments  is  greater  than  the  sum 
of  the  2d  and  3d  interests,  form  the  amount  and  subtract  therefrom 
the  sum  of  the  2d  and  3d  payments.  The  difference  is  the  4th  prin- 
cipal.    Find  the  4th  interest. 

Since  the  4th  payment  is  greater  than  the  4th  interest,  form  the 
amount  and  subtract  therefrom  the  4th  payment.  Compute  the  in- 
terest on  the  diflference,  the  5th  principal,  to  the  last  date,  and  form 
the  amount,  which  is  the  sum  then  due. 

Notes. — 1.  Sometimes  an  estimate  of  the  interest  may  be  made  men- 
tally with  sufficient  accuracy  to  determine  Avhether  it  is  greater  or  less 
than  the  payment.  If  greater,  the  sum  of  the  two  or  more  decimal 
multipliers,  can  be  used  for  a  multiplier.  Instead  of  multiplying 
by  .018  and  .012  above,  their  sum,  or  .03,  might  have  been  used. 
When  the  rate  is  other  than  6%,  the  several  interests  should  be  in- 
creased or  diminished,  as  the  rate  may  require,  before  forming  the 
amounts. 

2.  The  above  rule  is  generally  used  when  the  time  between  the 
date  of  the  note  and  its  payment  exceeds  one  year. 

2.  A  note  of  $600,  dated  June  10,  1867,  had  indorse- 
ments as  follows:  Dec.  4,  1867,  $50;  Mch.  25,  1868,  $12; 
July  9,  1868,  $75.  How  much  was  due  Oct.  15,  1868,  at 
6  %  interest  ? 

3.  A  note  of  $1000,  dated  Apr.  10,  1864,  was  indorsed 
as  follows:  Nov.  10,  1865,  $80.50;  July  5,  1866,  $100; 
Jan.  10,  1867,  $450.80;  Oct.  1,  1869,  $500.  What  was 
due  Jan.  1,  1870,  at  7  %  interest? 

4.  A  note  of  $450,  dated  July  4,  1868,  was  indorsed  as 
follows :  Jan.  20,  1869,  $15 ;  June  9,  1869,  $200 ;  Oct.  20, 
1869,  $10.     What  was  due  Jan.  10,  1870,  at  10  %  interest? 

5.  A   note  of -$850,  dated  March  4,  1865,  had  indorse- 


184  COMPLETE  ARITHMETIC. 

ments  as  follows:  Sept.  1,  1865,  $12;  May  4,  1866,  $10; 
Sept.  15,  1866,  $250 ;  Jan.  20,  1867,  $400.  What  was  due 
July  1,  1868,  at  6  %  interest? 

6.  A  note  of  $520,  dated  Apr.  12,  1867,  had  three  in- 
dorsements as  follows:  Dec.  6,  1867,  $120;  July  9,  1868, 
$12  ;  Nov.  30,  1868,  $9.  What  was  due  May  1,  1869,  at 
9  %  interest? 

7.  $1250.  Cincinnati,  July  1,  1868. 

On  demand,  I  promise  to  pay  Peter  Smith,  or  order, 
twelve  hundred  and  fifty  dollars,  with  interest  at  7^%,  for 
value  received.  John  Coons. 

Indorsements:  Sept.  14,  1868,  $300;  Jan.  20,  1869,  $12; 
Oct.   20,    1869,  $20;    Nov,   8,  1869,  $500. 

What  was  due  on  the  above  note  Jan.  1,  1870? 

8.  $1000.  San  Francisco,  Apr.  10,  1867. 
For  value  received,  I  promise  to  pay  to  Wm.  Penn,  Jr., 

or  order,  thirty  daj^s  after  date,  one  thousand  dollars,  with 
interest  at  10  %.  Gould  Dives. 

IndorsemenU:  July  28,  1877,  $500;  Dec.  13,  1867,  $8; 
Feb.  25,  1868,  $12;  July  7,  1868,  $125;  Oct.  3,  1868, 
$200;  Mch.  15,  1869,  $50. 

What  was  due  on  the  above  note  June  3,  1869. 

294.  When  partial  payments  are  made*  on  mercantile  ac- 
counts, past  due,  and  on  notes  running  a  year  or  less,  the 
interest  is  often  computed  by 

THE  MERCHANT'S  RULE. 

Compute  the  interest  on  the  principal  from  the  time  it  begins  to 
draw  interest  to  the  time  of  settlement,  and  also  on  each  payment 
from  the  time  it  was  made  to  the  time  of  settlement. 

From  the  sum  of  the  principal  and  its  interest,  subtract  the 
sum  of  tJie  payments  and  their  interests,  and  the  difference  ivill 
he  the  balance  due. 


INTEREST.  18^ 

9.  A  note  of  $800,  dated  March  12,  1869,  and  drawing 
interest  at  8  %,  was  indorsed  as  follows:  May  15,  1869, 
$200;  Aug.  10,  1869,  $75;  Oct.  20,  1869,  $125.  What 
was  due  Dec.  30,  1869? 

10.  Payments  were  made  on  a  debt  of  $350,  due  Feb.  1, 
1868,  as  follows:  March  20,  1868,  $45;  May  1,  1868,  $60; 
July  5,  1868,  $80 ;  Oct.  1,  1868,  $50.  What  was  due  Nov. 
1,  1868,  at  6%.  interest? 

Notes. — 1.  There  are  several  other  rules  for  computing  interest 
when  partial  payments  are  made,  but  they  are  not  in  general  use, 
and  hence  are  omitted.  The  old  rules, called  the  "Vermont  Kule  " 
and  the  "  Connecticut  Kule,"  and  found  in  most  arithmetics,  have 
been  modified  by  recent  legislation. 

2.  The  chief  aim  of  legislative  enactments  on  this  subject  has  been 
to  protect  the  debtor  from  paying  interest  on  interest,  but  there  is  no 
essential  difference  between  applying  payments  to  the  discharge  of 
interest  instead  of  principal,  and  paying  interest  on  such  accrued  in- 
terest. The  debtor  loses  the  use  of  so  much  of  every  payment  as  is 
applied  to  interest,  and  the  creditor  gains  the  use  of  it.  The 
"  Merchant's  Eule  "  is  the  only  one  that  does  not  practically  allow 
interest  on  interest. 


FIVE  PROBLEMS  IN  INTEREST. 

295.  Five  quantities  are  considered  in  interest,  and  such 
is  the  relation  between  them  that,  if  any  three  are  given, 
the  other  two  may  be  found.  These  quantities  are  the 
Principal,  Bate  Per  Cent,  Time,  Interest,  and  Amount 
There  are  five  classes  of  problems  of  practical  importance. 

Problem  I. 

Principal,  Rate  Per  Cent.,  and  Time  given,  to  find 
tlie   Interest  and   Anaonnt. 

Note. — This  problem  has  already  been  considered.  The  follow- 
ing problems  may  be  solved  by  the  pupil  by  either  of  the  preceding 
methods,  but  the  time  in  the  last  three  problems  should  be  found  by 
the  method  by  days. 

1.  What  is  the  interest  of  $12.50  for  3  yr.  1  mo.  15  da., 
at6%?     Atia%? 

C.Ar.— la. 


186  COMPLETE  ARITHMETIC. 

2.  What  is  the  interest  of  $160.80  for  2  yr.  3  mo.  3  da., 
at7%?    At9%? 

3.  What  is  the  interest  of  $56.40  for  21  days,  at  10  %  ? 
What  is  the  amount  ? 

4.  What  is  the  interest  of  $1000  from  May  13,  1867,  to 
July  8,  1868,  at  6  %  ?     At  7^%? 

5.  What  is  the  amount  of  $204.50  from  Jan.  21,  1869, 
to  Feb.  3,  1870,  at  9  %  ?     At  12  %  ? 

6.  What  is  the  interest  of  $80.25  from  May  15,  1869, 
to  Sept.  24,  1869,  at  6  %  ?    At  8  %  ? 

7.  A  note  of  $920,  dated  Nov.  12,  1869,  was  paid  Apr. 
3,  1870:  what  was  the  amount,  at  9  %  ? 

8.  A  note  of  $7.50,  dated  Apr.  20,  1870,  was  due  Oct. 
12,  1870,  with  interest  at  8  %  :  what  was  the  amount  ? 

296.   Formulas. — 1.  Interest  =principal  X  rate%  X  time, 
2.  Amount=principal-^  interest. 

Problem  II. 

Principal,    Interest,    and.    Time    given,    to   find,    tlie 
Rate  I>er  Cent. 

9.  The  interest  on  $540  for  8  mo.  18  da.  was  $27.09: 
what  was  the  rate  per  cent.  ? 

Process. 

^540  Since  the  interest  on  $540  for  8   mo. 

.043  18  da.,  at  1%,  is  $3.87,  the  rate  %  which 

1620  produced    $27.09    interest,    was    as    many 

2160  times  1%    as  $3.87  is  contained  times  in 

6  )  $23.220   InLatQfo.  $27.09,  which  is   7.      Hence,  the  interest 

$3.87       "     "1%.  accrued  at  7%. 

$27.09  ^$3.87  =  7. 

Note. — The  interest  at  1%  may  be  found  bv  multiplying  $540  by 
^  of  .043,  whicli  is  .0071. 

10.  The   interest  of  $456   for   3   yr.    5  mo.    18  da.,   is 
$79.04:   what  is  the  rate  per  cent.? 

11.  The   interest  of  $216   for   5   yr.    7   mo.    27   da.,    is 
$122.22:  what  is  the  rate  per  cent.? 


PROBLEMS  IN  INTEREST.  187 

12.  The  interest  of  $560  for  2  yr.  4  mo.  15  da.,  was 
$106.40:  what  was  the  rate  per  cent.? 

13.  The  interest  of  $95.40  for  3  yr.  9  mo.,  is  $28.62: 
what  is  the  rate  per  cent.? 

14.  The  interest  of  $240  from  Feb.  15,  1868,  to  Apr. 
27,  1869,  was  $23.04:  what  was  the  rate  per  cent.? 

15.  The  interest  of  $252  from  Aug.  2,  1867,  to  March  9, 
1868,  was  $12,152:  what  was  the  rate  per  cent.? 

16.  A  note  of  $345.60,  dated  Feb.  5,  1863,  was  paid 
Aug.  20,  1865,  and  the  amount  was  $407,088:  what  was 
the  rate  per  cent.? 

FORMULA  AND  RULK 

297.  Formula. — Rate  %  =  interest  -i-  {prin.  X  1  %  X  time). 

298.  KuLE. — To  find  the  rate  per  cent..  Divide  the  given 
interest  by  the  interest  of  the  principal  for  the  given  time,  at  1 
per  cent. 

Problem  III. 

I'rin.cipal,   Interest,   and    Rate  IPer  Cent,  given,  to 
find   the  Time. 

17.  The  interest  of  $300,  at  9%,  is  $60.75:  what  was 
the  time? 

Process.  Since  the  interest  of  $300  for  1  year, 

^3QQ  at  9%,  is  $27,  $300  must  be  on  interest 

"             ,09  as  many  years  to  produce  $60.75  inter- 

^27.00     Int.  for  1  yr.  ^^^y  ^^  ^^7  are  contained  times  in  $60.75, 

which  is  2.25.     Hence  the  time  is  2.25 

$60.75  -^  $27  =  2.25  years,  or  2  yr.  3  mo.  (Art.  170). 

2.25  yr.  =  2  yr.  3  mo. 

18.  The  interest  of  $908,  at  3|  %,  was  $79.45:  what  was 
the  time? 

19.  The  interest  of  $56.78  for  a  certain  time,  at  10%, 
was  $22.23:  what  was  the  time? 

20.  How  long  must  a  note  of  $300  run  to  give  an  amount 
of  $347.25,  at  6  %  ? 


188  COMPLETE  ARITHMETIC. 

21.  In  what  time  will  the  interest  of  $150,  at  4%,  be 
$9?     Sll? 

22.  In  what  time  will  $2040  produce  $334.05  interest,  at 
5%? 

23.  In  what  time  will  any  principal  double  itself  at  4  % 
interest?     At  6  %  ?     At  10  %  ? 

24.  In  what  time  will  any  principal  double  itself  at  5  % 
interest?     At  7  %  ?     At  12  %  ? 

25.  In  what  time  will  any  principal  treble  itself,  at  5  % 
interest  ?     At  10  %  ?     At  6  %  ? 

FORMULA  AND  RULE. 

299.  Formula. — Time  =  interest  -=-  (^principal  X  rate  %). 

300.  Rule. — To  find  the  time.  Divide  the  given  interest  by 
the  interest  of  the  'principal  for  1  year,  at  the  given  rate  per 
cent. 

Note. — Reduce  the  fraction  of  a  year  to  months  and  days.  If 
preferred,  the  interest  may  be  divided  by  the  interest  for  1  month,  at 
the  given  rate  per  cent. 

Problem  IV. 

Interest,  Rate    3?er  Cent.,   and  Time   given,  to  find 
the   Principal. 

26.  What  principal  will  produce  $49.20  of  interest  in  1 
yr.  4  mo.  12  da.,  at  6  %  ? 

1st  Process.  Since  $1  of  principal  produces 

$.082=Int.  of  $1.  ^-082   of  interest,  it   will   take  as 

many  times  %1  of  principal  to  pro- 
$.082  )  $49.20  (  600  ^^uce  $49.20  of  interest  as  $.082  is 
492 
contained  times  in  $49.20,  which  is 

$1  X  600  =  $600.     Ans.       600.     600  times  $1  =  $600,  the  re- 
quired principal.     Or, 

2d  Process.  Since  the  interest  of  $1  is  .082 

.082 )  $49,200  (  $600  of  itself,  $49.20  is  .082  of  the  re- 

492  quired  principal.    S49.20h-  .082  = 

$600. 


PROBLEMS  IN  INTEREST.  189 

What  principal  will  produce 

27.  $15.24  interest  in  7  mo.  6  da.,  at  8  %  ? 

28.  $1000  interest  in  5  yr.  6  mo.   20  da.,  at  5%? 

29.  $519  interest  in  5  mo.  23  da.,  at  12  %? 

30.  What  sum  invested,  at  7%,  will  produce  $378  inter- 
est annually  ? 

31.  What  sum  invested,  at  4|^  %,  will  yield  an  annual 
income  of  $900? 

32.  AVhat  principal  will  produce  $220  interest,  from  Oct. 
25,  1871,  to  March  7,  1872,  at  8  %  ? 

33.  What  principal  will  produce  $17.78  interest,  from 
Jan.  10,  1872,  to  March  13,  1872,  computed  by  days,  at  4%? 

FORMULA  AND  RULES. 

301.  Formula. — Principal  =  interest -^  (rate  %  X  time). 

302.  Rules. — To  find  the  principal  when  the  interest,  rate 
per  cent.,  and  time  are  given,  1.  Divide  the  given  interest  by 
the  interest  of  $1  for  the  given  time  and  rate  per  cent.,  and 
multiply  $1  by  the  quotient.     Or, 

2.  Divide  the  given  interest  by  the  decimal  corresponding  to 
the  interest  of  $1  for  the  given  time  and  rate  per  cent. 

Problem  V. 

A-iTionnt,  R-ate   l^er  Cent.,  and   Xiine   given,    to  find 
the    Principal. 

34.  What  principal,  on  interest  at  8%,  for  1  yr.  6  mo. 
18  da.,  will  give  an  amount  of  $730.60? 

The  interest  of  $1  for  1  yr.  6  mo. 

18  da.  is  $.124,  and  the  amount  is 

1ST  FROCEss.  ^j^24.     If   the   amount   of  $1   is 

$1,124  =  amount  of  $1.  $1,124,  it  will  take  as  many  times 

S1.124  )  $730,600  (  650.  $1  to  yield  an   amount  of  $730.60 

^ZM_  as   $1,124    is    contained    times    in 

§???  $730.60,  which   is   650.     650  times 

$1  is  $650,  the  required   principal. 

Or, 


5620 
0 


190  COMPLETE   ARITHMETIC. 

2d  Process. 

1.124  )  $730,600  (  $650.  ^^"^^  ^^^  ^°^^"^*  «^  ^^  ^^  1-124 

6744  of  itself,  $730.60  is  1.124  of  tlie  re- 

~5620  quired  principal.    $730.60-4-1.124 

6620  =  $650. 

35.  What  principal  on  interest,  at  5  %,  for  1  yr.  10  mo. 
12  da.,  will  amount  to  $70.24? 

36.  What  sum  of  money  put  at  interest,  at  7  %,  for  8 
mo.  18  da.,  will  amount  to  $567.09? 

37.  What  sum  of  money  put  at  interest,  at  8  %,  for  2 
yr.  1  mo.  15  da.,  will  amount  to  $421.20? 

38.  What  sum  of  money  put  at  interest  March  15,  1870, 
at  6  %,  will  amount  to  $2600.40,  Aug.  6,  1871? 

FORMULA  AND  RULE. 

303.  Formula. — Principal  =  amt.-T-ll  -}-  (rate  %  X  iime)\. 

304.  Rules. — 1.  To  find  the  principal  when  the  amount, 
rate  per  cent.,  and  time  are  given,  1.  Divide  Uie  given  amount 
by  the  amount  of  $1  for  the  given  time  and  rate  per  cent.j  and 
multiply  $1  by  the  quotient.     Or, 

2.  Divide  the  given  amount  by  ike  decimal  corresponding  to 
the  amount  of  $1  for  the  given  time  and  rate  per  cent. 

Note. — The  principal  thus  found  is  the  present  worth  of  the  amount. 
See  Art.  306. 


REVIEW  OF  THE  FIVE  PROBLEMS. 

305.  The   formulas  for  the  five  problems  in  interest  are 
here  presented  together : 

Formulas. — 1.  Interest  =^  principal  X  fate  %  X  time. 

2.  Eate%  =  int.  —-  (principal  X  1%  X  time) 

3.  Time  =  interest  -r-  (principal  X  'rate  % ). 

4.  Principal ::=  interest  -h-  (rate  %  X  time). 

5.  Principal = amt -i- [I -\-(rat€%  Xtinw)']. 


PROBLEMS  IN  INTEREST.  191 


WRITTEN   PROBLEMS. 


1.  What  is  the  interest  of  $205  for  2  yr.  5  mo.  24  da., 
at  7%? 

2.  What  is  the  amount  of  $160,  from  Jan.  12,  1869,  to 
July  3,  1870,  at  8  %  ? 

3.  At  what  rate  per  cent,  will  $512.60  yield  $25.72  in- 
terest in  8  mo.  18  da.  ? 

4.  The  principal  is  $126.75,  the  interest  $20.96,  and  the 
time  2  yr.  24  da. :  what  is  the  rate  ? 

5.  How  long  will  it  take  $5000  to  produce  $1125  inter- 
est, at  8  %  ? 

6.  The  principal  is  $326.50,  the  interest  $2.76,  and  the 
rate  8  %  :  what  is  the  time  ? 

7.  The  amount  is  $1563.75,  the  interest  $63.75,  and  the 
rate  7|-  %  :  what  is  the  time  ? 

8.  What  principal  will  yield  $1.36  interest  in  20  days, 
at  6%? 

9.  The  interest  on  a  certain  principal  from  Nov.  11, 
1857,  to  Dec.  15,  1859,  at  6%,  was  $4,474:  what  was  the 
principal  ? 

10.  What  principal,  at  7  %,  will  amount  to  $659.40  in  8 
months  ? 

11.  The  interest  is  $12.78,  the  time  1  yr.  2  mo.   6  da., 
and  the  rate  6  %  :  what  is  the  amount  ? 

12.  What  principal  will  amount  to  $609.20  in  4  mo.  18 
da.,  at  4%? 

13.  What  principal  will   amount  to  $288.85  in  1  yr.   6 
mo.,  at  6  %? 

14.  What  principal  will   produce  $21,757  interest,  from 
Jan.  1.  to  Oct.  20,  1869,  at  7  %  ? 

15.  How  long  will  it  take  any  principal  to  double  itself 
at  6  %  ?     At  4  %  ?     At  10  %  ? 

16.  What  is  the  amount  of  $420,  from  June  10,  1869, 
to  Jan.  21,  1870,  at  lO-^? 

17.  What  sum,  bearing  interest  at  7%,  will  yield  an  an- 
nual income  of  $1000? 


19^  COMPLETE  ARITHMETIC. 


PRESENT  WORTH  AND  DISCOUNT. 

306.  The  most  common  application  of  Problem  V  is  In 
computing  Present  Worth  and  Discount. 

307.  The  JPresent  Worth  of  a  debt  due  at  a  future 
time,  without  interest,  is  the  sum  or  principal  which,  at  Ihe 
current  rate  of  interest,  will  amount  to  the  debt  when  it 
becomes  due. 

308.  Disco U^lt  is  the  amount  deducted  from  a  debt 
for  its  payment  before  it  is  due. 

309.  True  Discount  is  the  difference  between  a  debt, 
not  bearing  interest,  and  its  present  worth. 

•        True  discount  is  the  interest  on  the  'present  woi^th  of  a  debt,  while 
simple  interest  is  computed  on  the  debt  itself.     The  difference  is  the 
.  interest  on  the  true  discount  for  the  time. 

WRITTEN  PROBLEMS. 

1.  What  is  the  present  worth  of  a  note  of  S212,  due  1 
year  hence,  without  interest,  the  current  rate  of  interest 
being  6  %  ?     What  is  the  true  discount  ? 

Process.  The  amount  of  $1  for  1  year, 

$1.06  )  $212.00  (  200  at  6%,   is  $1.06,  and   hence   the 

212  present  worth  of  $1.06,  due  1  year 

$1  X  200  =  f2^  Present  .oorth.       >■;" <•«' '«  *\^^[l^'  P''^™'  ^"'f' 

$212- 200  =  $12,  True  discount.     "{  f-^^  ^  »'  "'«  P^!^™'  Zf^ 
'  of  $212.  IS  as  many  times  $1  as 

$1.06  is  contained  times  in  $212,  which   is  200.     Or,  since  $1.06  is 

1.06  of  $1,  $212  is  1.06  of  its  present  worth.     $212  ~  1.06  =$200. 

Note. — This  is  only  an  application  of  Problem  V.,  the  debt  being 
the  amount,  the  present  worth  the  principal,  and  the  true  discount  the 
interest. 

2.  What  is  the  present  worth  of  a  bill  of  $260  due  in  8 
months,  without  interest,  the  current  rate  of  interest  being 
6 % ?     What  is  the  true  discount? 


PRESENT  WORTH  AND  DISCOUNT.  193 

Find  the  present  worth  and  the  true  discount  of 

3.  $220  due  in  1  yr.   6  rao.,  without  interest,  current 
rate  1%. 

4.  $145.60  due  in  8  mo.   12  da.,  without  interest,  cur- 
rent rate  8  % . 

5.  $305.75  due  in  9  mo.  6  da.,  without  interest,  current 
rate  9  %. 

6.  $1250  due  in  1  yr.  7  mo.  21  da.,  without  interest, 
current  rate  5  % . 

7.  $1508  due  in  90  days,  without  interest,  current  rate 
7%. 

8.  $2040.50  due  in  36  days,  without  interest,  current 
rate  10%. 

9.  $884,125  due   in   1   yr.  2  mo.  10  da.,  without  inter- 
est, current  rate  6  % . 

10.  What  is  the  difference  between  the  true  discount  and 
the  interest  of  $216  for  2  years,  at  8  %  ? 

What  is  the  difference  between  the  true  discount  and  the 
interest  of 

11.  $199.80  for  1  yr.  10  mo.,  at  6%  ?     At  12  %  ? 

12.  $666.40  for  2  yr.  4  mo.  15  da.,  at  6  %  ?     At  8  %  ? 

13.  $534  for  1  yr.  1  mo.  18  da.,  at  6  %  ?     At  5  %  ? 

14.  $175.20  for  1  yr.  10  mo.  12  da.,  at  9  %  ? 

15.  $1250.60  for  1  yr.  7  mo.  24  da.,  at  5  %  ? 

16.  $884.12  for  96  days,  at  7|-  %  ?     At  10  %  ? 

17.  How  large  a  note,  due  in  1  yr.  6  mo.,  with  interest 
at  7  %,  will  cancel  a  debt  of  ^$442,  due  in  1  yr.  6  mo.,  with- 
out interest? 

18.  What  is  the  difference  in  the  present  value  of  a  cash 
payment  of  $345,  and  a  note  of  $371  due  in  9  months,  with- 
out interest,  the  use  of  money  being  worth  8  %  ? 

FORMULAS  AND  RULES. 

310.  Formulas. — 1.  Present  worth ^^ debt —[1  +  (rate  % 
X  time)^. 
2.   True  discount  =  debt  —  present  worth. 

C.Ar.— 17. 


194 


COMPLETE  ARITHMETIC. 


311.  Rules. — 1.  To  find  the  present  worth  of  a  debt  due 
at  a  future  time,  without  interest,  1.  Divide  the  debt  hj  tJie 
amount  of  $1  for  the  given  time,  at  the  current  rate  of  interest, 
and  multiply  $1  by  the  quotient.     Or, 

2.  Divide  the  debt  by  the  decimal  corresponding  to  the  amount 
of  $1  for  Hie  given  time,  and  at  the  current  rate  of  interest. 

BANK  DISCOUNT. 


312.  ^anh   J)iscouut  is 


the  interest  on  a  note  for 
the  number  of  days  from 
the  time  it  is  discounted  to 
the  time  it  is  legally  due. 

313.  TJie  Proceeds 

of  a  note  are  its  face,  or 
the  sum  discounted,  less 
the  discount.  The  pro- 
ceeds are  also  called  Avails 
and  Cash  Value. 

314.  Days  of  Grace 

are  the  three  days  allowed 
for  the  payment  of  a  note 
after  the  specified  time 
has  expired. 

Note. — A  note  is  payable,  or  nominally  due,  at  the  expiration  of  the 
specified  time,  but  it  does  not  mature,  or  become  legalbj  due,  until  the 
last  day  of  grace,  or  the  day  preceding,  when  the  last  day  of  grace 
falls  on  Sunday  or  a  legal  holiday.  The  date  of  expiration  and  the 
date  of  maturity  are  usually  written  with  a  line  between  them ;  thus, 
.Fan.  9  /  ]2. 

315.  When  a  bank  loans  money,  the  borrower  gives  his 
note  payable  at  a  specified  time,  without  interest.  This  note 
is  then  discounted  by  the  bank  for  the  actual  number  of 
days  in  the  time  plus  the  three  days  of  grace,  and  the  pro-^ 
ceeds  are  paid  to  the  borrower. 

316.  When  a  note  drawing  interest  is   discounted  by  a 


BANK  DISCOUNT.  195 

bank,  the  discount  is  computed  on  the  amount  of  the  note 
at  the  time  of  its  maturity. 

Business  men  generally  discount  notes  and  bills,  not  drawing  in- 
terest, by  deducting  the  interest  for  the  time,  with  or  without  grace 
as  per  agreement.  This  is  sometimes  called  Business  Discount.  The 
rate  of  interest  allowed  is  usually  greater  than  the  current  rate. 

Bills  due  in  three,  four,  or  six  months  are  often  discounted  by  de- 
ducting 5%  or  more  of  their  face,  without  regard  to  time.  This  is 
called  Per  Cent.  Off. 

WRITTEN  PROBLEMS. 

1.  What  is  the  bank  discount  of  a  note  of  S350,  payable 
in  60  days,  discounted  at  10  ^  ?     What  are  the  proceeds? 

Process. 
$350. 
.0105 

1750         Time  =60  da.  +  3  da.  =  63  da. 
3  50 


6  )  $3.6750 

$.6125  X  10  =  $6,125,  Bank  discount. 
$350  —  $6,125  =  $343,875,  Proceeds. 

2.  What  is  the  bank  discount  of  a  note  of  $250,  payable 
in  90  days,  discounted  at  8  %  ?     What  are  the  proceeds  ? 

Find  the  bank  discount  and  the  proceeds  of  a  note  of 

3.  $145,  payable  in  60  days,  discounted  at  7%. 

4.  $80.50,  payable  in  30  days,  discounted  at  8  %. 

5.  $1000,  payable  in  90  days,  discounted  at  7^%. 

6.  $750,  payable  in  45  days,  discounted  at  9  %. 

7.  $1250,  payable  in  100  days,  discounted  at  6%. 

8.  $56,  dated  Jan.   1,  1870,  payable  May  1,  1870,  dis- 
count 6%. 

9.  $120,  dated  Apr.  3,  1869,  payable  June  15,  1869,  dis. 
count  8  %. 

10.  $500,  dated  Dec.   15,   1870,  payable  Feb.  18,  1871, 
discount  9  %. 

11.  $8.75,  dated  Nov.  21,  1870,  payable  Mch.  12,  1871, 
discount  9  %. 


196  COxMPLETE  ARITHMETIC. 

12.  $400,  dated  Mch.  4,  1871,  payable  July  24,  1871, 
discount  5  %  ? 

13.  What  is  the  difference  between  the  bank  discount 
and  the  true  discount  of  $1319.50,  due  in  90  days,  dis- 
counted at  6  %  ? 

14.  What  is  the  difference  between  the  bank  discount 
and  the  true  discount  of  $768,  due  in  108  days,  discount 
8%? 

15.  What  is  the  difference  between  the  bank  discount 
and  the  true  discount  of  $3330.80,  due  in  45  days,  dis- 
count 7%? 

16.  A  note,  dated  Apr.  10,  1871,  is  payable  in  90  days: 
what  is  the  time  of  its  maturity? 

Note. — The  date  of  maturity  is  found  by  counting  forward  the 
number  of  days  phis  three  days,  when  the  time  is  expressed  in  days ; 
and  the  number  of  calendar  months  plus  three  days,  when  the  time 
is  expressed  in  months. 

17.  A  note,  dated  Feb.  6,  1868,  was  payable  in  60  days: 
what  was  the  date  of  its  maturity  ? 

18.  A  note,  dated  Aug.  9,  1870,  is  payable  4  months 
from  date :  what  is  the  date  of  its  maturity  ? 

19.  A  note  of  $460,  dated  Apr.  3,  1870,  and  payable  in 
90  days,  with  interest  at  6  %,  was  discounted  May  10,  1870, 
at  8  %  :  what  were  the  proceeds  ? 

Suggestion.— Find  the  amount  of  $460  for  93  days,  at  6%,  and 
then  discount  this  amount  for  56  days,  at  8%. 

20.  A  note  of  $125,  dated  May  21,  1870,  and  payable  in 
60  days,  with  interest  at  6  %,  was  discounted  May  25,  1870, 
at  10  %  :  what  were  the  proceeds  ? 

21.  A  note  of  $1000,  dated  Aug.  15,  1869,  and  payable 
in  6  months,  with  interest  at  7%,  was  discounted  Nov.  27, 
1869,  at  9  % :  what  was  the  bank  discount? 

Suggestion. — Compute  the  interest  for  6  mo.  3  da.,  and  the  dis- 
count for  83  days. 

22.  A  note  of  $90,  dated  Apr.  12,  1870,  and  payable  in 


BANK  DISCOUNT.  197 

4  months,  with  interest  at  5  %,  was   discounted  June  1, 
1870,  at  7  %  :  what  were  the  proceeds? 

23.  A  note  of  $650,  dated  Mch.  2,  1869,  payable  Apr. 
1,  1870,  and  indorsed  $300  Oct.  1,  1869,  was  discounted 
Feb.  3,  1870:  what  were  the  proceeds? 

24.  For  what  sum  must  a  note,  payable  in  60  days,  be 
drawn  to  produce  $493,  when  discounted  at  8  %  ? 

Since  the  proceeds  of  $1  are 

Process.  $.986,  it  will  require  as  many 

$1  —  $.014  =  $.986,  Proceeds  of  $1.      times  $1   to  produce   $493  as 

$493  -^  $.986  =::  500  $.986  is  contained  times  in  $493, 

$1  X  500  =  $500,  Face  of  note.  which  is  500 ;  and  500  times  $1 

=  $500. 

25.  For  what  sum  must  a  note,  payable  in  90  days,  be 
drawn  to  produce  $1969,  when  discounted  ai  6  %  ? 

26.  What  must  be  the  face  of  a  note,  dated  July  5,  1871, 
and  payable  in  4  months,  to  produce  $811,  when  discounted 
at  9  %  ? 

27.  What  must  be  the  face  of  a  note,  dated  Jan.  10, 
1870,  and  payable  in  3  months,  to  produce  $1958,  when 
discounted  at  12%? 

28.  A  merchant  discounted  a  bill  of  $750,  payable  in  4 
months,  by  deducting  the  interest  for  the  time  without 
grace,  at  10  %  :    what  were  the  cash  proceeds  of  the  bill  ? 

29.  A  note  of  $340,  due  in  9  months,  without  interest, 
was  discounted  by  deducting  the  interest  for  the  time,  at 
8  % :  what  was  the  cash  value  of  the  note  ? 

30.  A  merchant  having  sold  a  bill  of  goods  amounting 
to  $1030,  on  three  months'  time,  allowed  5  %  off  for  cash: 
what  were  the  cash  proceeds  of  the  sale? 

31.  A  retail  dealer  having  bought  $950  worth  of  goods, 
on  6  months'  time,  cashed  the  bill  for  7^%  off:  what  were 
the  cash  proceeds? 

32.  A  merchant  bought,  March  20,  1870,  a  bill  of  goods 
amounting  to  $3540,  on  three  months'  time,  but,  being 
offered  5  %  off  for  cash,  he  borrowed  the  money  at  a  bank 


198  COMPLETE  ARITHMETIC. 

for  the  time,  at  10%,  and  cashed  the  bill.     How  much  did 
he  g^in  by  the  transaction? 

FORMULAS  AND  RULES. 

317.  Formulas. — 1.  Bank  discount  =  sum  discounted  X 
(int.  of  $1  for  the  days  -j-  3  days). 

2.  Business  discount  =  sum  discounted  y, 
int.  of  $1  for  the  time. 

3.  Proceeds  =  sum  discounted — discount. 

318-  EuLES. — 1.  To  compute  bank  discount,  Find  the  in- 
terest on  the  sum  discounted  at  the  given  rate  per  cent,  and  for 
the  actual  number  of  days  in  the  time  plus  three  days. 

2.  To  compute  business  discount,  Find  the  interest  on  the 
su7n  discounted,  at  the  given  rate  per  cent,  and  for  the  given 
time. 

3.  To  find  the  proceeds,  Subtract  the  discount  from  the  sum 
discounted. 

4.  To  find  the  face  of  a  note  to  yield  given  proceeds. 
Divide  the  given  proceeds  by  $1  minus  tfw  interest  of  $1  for  the 
given  time  ivith  grace,  and  midtiply  $1  by  the  quotient. 

Notes. — 1.  In  discounting  a  note  bearing  interest,  the  interest  is 
computed  by  months  or  by  days,  according  as  the  time  is  expressed 
in  the  note,  but  the  discount  is  usually  computed  by  days. 

2.  Business  discount  is  computed  by  months  or  days,  according  as 
the  time  is  expressed  in  the  paper  discounted,  and  with  or  without 
grace.     In  these  respects  it  differs  from  bank  discount. 

3.  Bank  discount  is  not  only  interest  paid  in  advance,  but  the  in- 
terest is  computed  on  both  the  proceeds  and  the  discount.  The  bor- 
rower pays  interest  on  more  money  than  he  receives. 

PROMISSORY  NOTES  AND  DRAFTS. 

I.  PROMISSORY  NOTES. 

319.  A  Promissory  Note  is  a  written  agreement  by 
one  party  to  pay  to  another  a  specified  sum  at  a  specified 
time.  The  sum  whose  payment  is  promised,  is  called  the 
Face  of  the  note. 

The  person  wlio  signs  a  note  is  called  its  Maker ;  the  person  to 
whom  it  is  payable  is  the  Payee;  and  its  owner  is  the  Holder. 


PROMISSORY  NOTES.  199 

320.  A  Johif  Note  is  a  note  signed  by  two  or  more 
persons  who  are  jointly  liable  for  its  payment. 

321.  A  Joint  and  Several  JV^ote  is  a  note  signed 
by  two  or  more  persons  who  are  both  jointly  and  singly 
liable  for  its  payment. 

322.  An  Indorser  is  a  person  who  signs  his  name  on 
the  back  of  a  note  as  security  for  its  payment. 

323.  The  following  are  the  more  common  forms  of  pro- 
missory notes  : 

FORM  I.— Demand  Note. 

$95jVo-  Nashville,  Tenn.,  May  1,  1870. 

For  value  received,  I  promise  to  pay  to  John  Wilson,  on 

demand,  Ninety-five  yW  Dollars. 

Henry  Smith. 

[  STAMP.] 

FORM  II.— Time  Note. 

$95tVo-  St.  Louis,  Mo.,  May  1,  1870. 

Ninety  days  after  date,  I  promise  to  pay  to  John  Wilson, 
or  bearer,  Ninetyfive  -f^  Dollars,  with  interest,  for  value  re- 
ceived. 

Henry  Smith. 

[  STAMP.] 


FORM  III. —  Joint  and  Several  Note. 

$95yVV-  Louisville,  Ky.,  March  12,  1870. 

Four  months  after  date,  we  jointly  and  severally  promise 
to  pay  Henry  Cooke,  or  order,  Ninety-five  y'^fy-  Dollars,  with  in- 
terest at  S%,  for  value  received. 

Thomas  Hughes, 

[  STAMP.] 

Charles  G.  Knight. 


200  COMPLETE  ARITHMETIC. 


FORM  IV. — Note  Payable  at  a  Bank. 

$500.  Baltimore,  Md.,  Apr.  10,  1870. 

Sixty  days  after  date,  ive  -promise  to  pay  to  Wilson,  HinUe 

&  Co.,  or  order,  at  the  First  National  Bank,  Five  Hundred 

Dollars,  for  value  received. 

Charles  Cooke  &  Co. 

[  STAMP.] 

Kemarks. — 1.  A  note  should  contain  the  words  "  value  received," 
otherwise  the  holder  may  be  required  to  prove  that  the  maker  re- 
ceived its  value. 

2.  When  the  time  for  the  payment  of  a  note  is  not  specified,  it  is 
due  on  demand.  If  the  place  of  payment  is  not  mentioned,  it  is  pay- 
able at  the  maker's  residence  or  place  of  business. 

3.  When  a  note  contains  the  words  "  with  interest,"  and  no  rate  is 
specified  (Form  II.),  interest  accrues  at  the  legal  rate.  If  the  words 
"  with  interest,"  are  omitted  (Form  IV.),  no  interest  accrues  until 
after  maturity,  when  the  note  draws  interest,  at  the  legal  rate,  until 
paid. 

324.  A  JSegotiable  Wote  is  one  which  may  be  bought 
and  sold. 

A  note  is  negotiable  when  it  is  made  payable  "  to  the  bearer,"  or 
to  the  payee  "  or  bearer,"  or  to  the  payee  "  or  order,"  or  "  to  the 
order  of"  the  payee.     A  note  drawn  as  in  Form  I.  is  not  negotiable. 

A  note  made  payable  to  the  bearer  is  negotiable  without  indorse- 
ment. U.  S.  treasury  notes  and  bank  notes,  used  as  money,  are  pay- 
able to  the  bearer,  and  are  transferred  by  delivery. 

A  note  payable  to  order  must  be  indorsed  by  the  payee  before  it  is 
negotiable. 

When  the  payee  indorses  a  note  by  simply  writing  his  name  on 
the  back,  it  is  called  an  indorsement  in  blank,  and  the  note  is  payable 
to  the  holder.  When  the  indorser  orders  the  payment  to  be  made  to 
a  particular  person,  as :  "  Pay  to  Charles  Williams,"  it  is  called  a 
special  indorsement. 

325.  If  the  maker  of  a  note  fails  to  pay  it  at  maturity, 
a  written  notice  of  the  fact,  made  by  a  notary  public,  is 
served  on  the  indorsers,  who  are  responsible  for  the  payment 
of  the  note.     Such  a  notice  is  called  a  Protest. 


BILLS  OF  EXCHANGE.  201 

Note.— A  protest  must  be  made  out  on  the  day  a  note  matures,  and 
■  it  must  be  sent  on  that  day  or  the  next,  otherwise  the  indorsers  are  not 
responsible. 

II.  DRAFTS,  OR  BILLS  OF  EXCHANGE. 

326.  A  Draft  is  an  order  made  by  one  person  upon 
another  to  pay  a  specified  sum  to  a  third  person  named. 
It  is  also  called  a  Bill  of  Exchange. 

The  person  who  makes  the  order,  is  called  the  Drawer  ;  the  person 
to  whom  it  is  addressed,  is  called  the  Drawee;  and  the  person  to 
whom  the  money  is  payable,  is  the  Payee. 

327.  The  following  are  the  common  forms  of  drafts: 

FORM  I.— Sight  Draft. 

$100.  Cincinnati,  O.,  Oct.  1,  1870. 

Pay  to  the  order  of  Bartlit  &  Smith,  One  Hundred 
Dollars,  and  place  to  the  account  of 

Charles  S.  Kelley. 
To  George  Brown,  Esq.,  New  York. 

[  STAMP.] 

FORM  II.— Time  Draft. 

$100.  Cincinnati,  O.,  Oct.  1,  1870. 

Thirty  days  after  sight  [or  date],  pay  to  the  order  of 
Bartlit  &  Smith,  One  Hundred  Dollars,  and  place  to  the  ac- 
count of 

Charles  S.  Kelley. 
To  George  Brown,  Esq.,  New  York. 

[  STAMP.] 

When  the  drawee  accepts  a  draft,  he  writes  the  word  "  Accepted," 
with  the  date,  across  the  face,  and  signs  his  name,  thus:  "Accepted, 
Oct.  3,  1870— Charles  S.  Kelley."  The  draft  is  then  called  an  Ac- 
ceptance, and  the  acceptor  is  responsible  for  its  payment. 

A  draft  made  payable  to  bearer  or  order  is  negotiable,  like  a  prom- 
issory note,  and  is  subject  to  protest,  in  case  payment  or  acceptance 
is  refused. 


202  COMPLETE  ARITHMETIC. 

Notes. — 1.  In  most  of  the  states  both  time  and  sight  drafts  are 
entitled  to  three  days  of  grace.  In  New  York  no  grace  is  allowed. 
on  sight  drafts. 

2.  When  a  draft  is  drawn,  ''acceptance  waived,"  it  is  not  subject  to 
protest  until  maturity  ;  and  when  an  indorser  writes  over  his  name, 
"demand  and  notice  waived,"  a  protest  in  his  case  is  not  necessary. 

3.  The  liability  of  an  indorser  of  a  note  or  draft  may  be  avoided 
by  his  writing  over  his  indorsement,  "  without  recourse." 

328.  A  Domestic  or  Inland  JBill  is  a  draft  which 
is  payable  in  the  country  where  it  is  drawn. 

A  Foreign  JSill  is  a  draft  which  is  drawn  in  one 
country  and  is  payable  in  another. 

329.  Exchange  is  the  process  of  making  payments  at 
distant  places  by  the  remittance  of  drafts,  instead  of  money. 

When  a  draft  can  be  bought  for  its  face,  it  is  said  to  be  at  'par; 
when  the  cost  is  less  than  the  face,  it  is  below  par,  or  at  a  discount; 
and  when  the  cost  is  more  than  the  face,  it  is  above  par,  or  at  a  pre- 
mium. The  rate  per  cent.,  which  the  cost  of  a  draft  is  more  or  less 
than  its  face,  is  called  the  Rate  of  Exchange. 

Notes. — 1.  The  rate  of  exchange  between  two  places  depends  chiefly 
on  their  relative  trade.  If  Cincinnati  owes  New  York,  drafts  on 
New  York  are  at  a  premium  in  Cincinnati ;  if  New  York  owes  Cin- 
cinnati, drafts  on  New  York  are  at  a  discount;  if  the  trade  of  the 
two  cities  with  each  other  is  equal,  exchange  is  at  par. 

2.  In  foreign  exchange,  drafts  are  expressed  in  the  currency  of 
the  country  on  which  they  are  drawn.  Tlie  comparative  value  of 
the  money  of  two  countries  is  called  the  Far  of  Exchange. 

AVRITTEN  PKOBLEMS. 

1.  What  is  the  cost  of  a  draft  on  New  York  for  $800, 
exchange  being  |  %  premium  ? 

Process  :  $800  X  -OOf  =  $6,  Prem.    $800  +  $6  =  $806,  Cost. 

2.  What  is  the  cost  of  a  draft  on  New  Orleans  for  $1250, 
at  ^  %  discount? 

3.  What  is  the  cost  of  a  draft  on  Philadelphia  for  $1050, 
at  J  %  premium  ? 

4.  A  merchant  in  St.  Louis  wishes  to  remit  $2500  by 
draft  to  New  York :  what  will  be  the  cost  of  the  draft,  ex- 
change being  1^  %  premium  ? 


EXCHANGE.  203 

5.  What  will  be  the  cost  of  a  draft  for  $500,  payable 
in  30  days  after  sight,  exchange  being  1  %  premium,  and 
interest  6  %  ? 

Process. 

$500. 

$500  X  -0055  =        $2.75  Discount  at  6%  for  33  days. 

$497.25  Proceed.^  of  Draft  (cost  at  par). 

$500  X  -01  =  MO  Premium  atl%. 

$502.25  Cost  of  Draft. 

Note. — If  preferred,  the  face  may  be  multiplied  by  the  cost  of 
$1,  which  is  $1  —  $.0055 -1- $.01,  or  $1.0045.  $500X1.0045  = 
$502.25. 

6.  What  will  be  the  cost  of  a  draft  for  $650,  payable  in 
60  days  after  sight,  exchange  being  |  %  premium,  and  in- 
terest 8  %  ? 

7.  What  will  be  the  cost  of  a  draft  for  $320,  payable  in 
45  days  after  sight,  exchange  being  f  %  discount,  and  in- 
terest 7  %  ? 

8.  How  large  a  sight  draft  can  be  bought  for  $259.52, 
exchange  being  If  %  premium  ? 

Process  :  $259.52  -=-  1.01 1  =  $256,     Face  of  Draft.     (Case  IV.) 

9.  How  large  a  sight  draft  can  be  bought  for  $962.85, 
exchange  being  If  %  discount? 

10.  A  sight  draft,  bought  at  ^  %  premium,  cost  $1256.25  : 
what  was  its  face? 

11.  How  large  a  draft,  payable  30  days  after  sight,  can 
be  bought  for  $502.25,  exchange  being  1  %,  and  interest  6  %  ? 

Process. 
$1  —  $.0055  =  $.9945      Proceeds  of  $1  discounted  for  33  days. 
$.9945  +  $.01  =$1.0045      Cost  of  $1. 
$502.25  -f-  $1 .0045  =  $500      Face  of  Draft. 

12.  How  large  a  draft,  payable  60  days  after  sight,  can 
be  bought  for  $798.80,  exchange  being  1^  %  premium,  and 
interest  8  %  ? 

13.  A  draft,  payable  in  30  days  after  sight,  was  bought 
for  $352.62,  exchange  being  1^  %  discount,  and  interest 
6  %  :  what  was  its  face? 


204  COMPLETE  ARITHMETIC. 


BONDS. 

330.  The  interest-bearing  notes  issued  by  nations,  states, 
cities,  railroad  companies,  and  other  corporations,  as  a 
means  of  borrowing  money,  are  called  Bonds. 

Bonds  are  issued  in  denominations  of  convenient  size,  with  interest 
usually  payable  annually  or  semi-annually,  and  they  are  made  nego- 
tiable like  certificates  of  capital  stock.     (Art.  243.) 

The  Coupons  attached  to  bonds  are  due-bills  for  the  interest,  which, 
as  the  interest  becomes  due,  are  cut  off  and  presented  for  payment. 

331.  The  several  classes  of  bonds  issued  by  the  United 
States  Government,  are  called  United  States  Securities^  or 
Government  Securities,  the  principal  of  which  are  known  as 
Sixes  of  1881,  Five -Twenties,  and  Ten-Forties. 

The  6's  of  '81,  bear  interest  at  6  %  in  gold,  payable  semi-annually, 
and  are  payable  in  1881. 

The  5-20's  bear  interest  at  6%  in  gold,  payable  semi-annually,  and 
are  redeemable,  at  the  option  of  the  Government,  after  five  years  from 
date,  and  are  payable  in  twenty  years.  The  several  series  were  issued 
in  1862  ( Old  5-20's),  1864,  1865,  1867,  and  1868. 

The  10-40's,  issued  in  1864,  bear  interest  at  5%  in  gold,  payable 
semi-annually  (except  the  $50  and  $100  bonds  bearing  annual  in- 
terest), and  are  redeemable,  at  the  option  of  the  Government,  in  ten 
years,  and  are  payable  in  forty  years. 

332.  The  market  value  of  United  States  bonds  is  quoted 
at  a  certain  per  cent,  of  their  par  value  or  face.  Bonds 
quoted  at  110  are  worth  in  currency  110%  of  their  face; 
that  is,  are  10  %  above  par.  The  quotation  includes  ac- 
crued interest. 

WRITTEN  PROBLEMS. 

1.  When   U.  S.  5-20's   are  quoted   at  109^,  what  will 
three  $500  bonds  cost  ? 

2.  When    U.  S.  6's  of  '81   are  worth    114,   Avhat  will 
$1250  in  bonds  cost? 

3.  A  widow  invested  $4725   in  U.  S.  10-40's,  at    105 : 
what  amount  in  bonds  did  she  receive  ? 


ANNUAL  INTEREST.  205 

4.  A  broker  invested  $26250  in  5-20's  at  1061   and  sold 
them  at  109 :   how  much  did  he  gain  ? 

5.  When  gold  is  at  115,  what  amount  in  currency  can 
be  bought  for  $8500  in  gold  ? 

6.  At  112,  what  amount  in  gold  can  be  bought  for  $1400 
in  treasury  notes  ? 

7.  When  gold  is  at  115,  and  U.  S.  10-40's  at  105,  what 
is  a  $500  bond  worth  in  gold  ? 

8.  When  gold  is  at  120  and  U.  S.  5-20's  at  115,  what  is 
the  gold  value  of  a  $1000  bond? 

9.  When  gold  is  at  110  and  U.  S.  10-40's  at  108,  what 
is  the  gold  value  of  a  $500  bond? 

10.  When  gold  is  worth  112|^,  what  is  the  gold  value  of 
a  dollar  treasury  note  ? 

11.  When  gold  is  at  115,  what  is  the  semi-annual  interest 
in  currency  on  $9500  in  10-40's  ? 

12.  When  gold  is  at  120,  what  rate  per  cent,  in  currency 
is  the  interest  on  5-20's? 

13.  A  college  invested  its  endowment  fund  in  U.  S.  5-20's, 
at  106 :  what  rate  of  interest  in  currency  will  it  receive 
when  gold  is  at  120  ?     At  110  ?     At  par  ? 


ANNUAL  INTEREST. 

333.  When  a  note  reads  "with  interest  payable  annually," 
the  interest  on  the  face  of  the  note,  due  at  the  close  of  each 
year,  is  called  Annual  Interest. 

334.  When  annual  interest  is  not  paid  at  the  close  of  the 
year,  when  due,  it  draws  simple  interest  until  paid. 

WRITTEN  PROBLEMS. 

1.  A  note  of  $500,  dated  May  10,  1870,  is  due  in  4  years, 
with  interest  at  6%,  payable  annually:  if  both  interest  and 
principal  remain  unpaid,  what  will  be  the  amount  due  on 
the  note  at  maturity? 


206  COMPLETE  ARITHMETIC. 

Process. 

$500  X  »06  =  S30,  Interest  on  principal  due  annually. 
$30  X  4  =     $120,  Total  annual  interest. 

$30  X  -06  X  3  =      5.40,  Simple  interest   on  1st  annual  interest  for  3  yrs. 
$30  X  .06  X  2  =     3.60,       "  "        "  2d       "  "        "  2   " 

$30  X  .06  X  1  =      1-80,       "  "        "  3c^       "  "        ^'1    " 

$130.80,  Ihtal  interest  due  at  maturity  of  note. 
500.00 


$630.80,  Amount  due  at  maturity  of  note. 

The  $30  annual  interest,  due  at  the  close  of  the  1st  year,  being 
unpaid,  draws  simple  interest  until  paid,  or  for  3  years;  the  $30 
annual  interest  due  at  the  close  of  the  2d  year,  draws  interest  for  2 
years ;  and  the  $30  annual  interest  due  at  the  close  of  the  3d  year, 
draws  interest  for  1  year.  The  fourth  annual  interest  is  paid  when 
due.  Hence,  the  total  interest  due  at  the  maturity  of  the  note,  con- 
sists of  (1)  the  annual  interest  for  1  year  ($30)  multiplied  by  4,  the 
number  of  years;  and  (2)  the  simple  interest  on  the  $30  annual  in- 
terest for  3  years,  2  years,  and  1  year,  or  for  6  years.  The  amount 
due  is  $500  +  $30  X  4  +  $1.80  X  6. 

2.  A  note  of  $750,  with  interest  payable  annually,  at  8%, 
was  paid  3  yr.  3  mo.  18  da.  after  date,  and  no  interest  had 
been  previously  paid :  what  was  the  amount  due  ? 

Process. 
$750  X  .OS  =    $60.       Interest  on  face  due  annually. 
$60  X  3.3  =  $198.00,  Total  annual  int.  for  3  yr.  3  mo.  18  da.  (3.3  yr.) 

$60  X  -312  =r  $18.72,  Simple  interest  on  %60for  3  ijr.  10  mo.  24  da. 

$750. 

).72,  Amount  doe. 


The  first  annual  interest  draws  simple  interest  for  2  yr.  3  mo. 
18  da. ;  the  second,  for  1  yr.  3  mo.  18  da. ;  and  the  third,  for  3  mo. 
18  da. ;  and  hence,  the  simple  interest  on  the  several  annual  interests 
equals  the  interest  of  $60  for  2  yr.  3  mo.  18  da.  +  1  yr-  3  mo.  18  da. 
+  3  mo.  18  da.,  or  for  3  yr.  10  mo.  24  da.  The  amount  due  consists 
of  (1)  the  principal;  (2)  the  total  annual  interest;  and  (3)  the  simple 
interest  on  the  annual  interest. 

3.  A  note  of  $1000,  with  annual  interest  at  6%,  is  due 
4  yr.  6  mo.  after  date :  no  interest  having  been  paid,  what 
will  be  due  at  maturity? 


ANNUAL  INTEREST.  207 

4.  A  man  bought  a  farm  for  $3500,  to  be  paid  in  4  years, 
with  interest  payable  annually,  but  failed  to  pay  the  inter- 
est: what  was  due  at  the  close  of  the  4th  year? 

5.  $650.  New  York,  July  1,  1869. 
On  the  first  day  of  January,  1872,  for  value  received,  I 

promise  to  pay  John  Black,  or  order,  six  hundred  and  fifty 
dollars,  with  annual  interest  at  7  % . 

Charles  Church. 

If  no  interest  be  paid  on  the  above  note,  what  will  be 
due  at  maturity? 

6.  If  the  above  note  and  interest  be  not  paid  until  Sept. 
12,  1872,  what  will  be  the  amount  due  ? 

7.  A  note  of  $800,  dated  March  18,  1867,  and  due  in  3 
years,  with  interest  at  6%,  payable  annually,  has  the  fol- 
lowing indorsements :  Oct.  24,  1868,  $150  :  Nov.  12,  1869, 
$240.     What  was  the  amount  due  March  18,  1870? 

Process. 
$800  X  .06  =  $48,  First  annual  interest. 

$48  X  2  =  $96,  Annual  interest  due  Mch.  18,  1869. 

$48  X  '06  ^=      4.80,     Interest  on  \st  annual  interest. 
SmSO,     Interest  due  Mch.  18,  1869. 
800 


§900.80,     Amount  due  Mch.  18,  1869. 

Payment,  $150.  |  ^  15360,     Amount  of  $150,  Mch.  18,  1869. 
Int.  on  same,  $3.60.  J      ^^^^;^^^     New  principal. 

$747.20  X  -06  =  $_44.832,  Interest  due  Mch.  18,  1870. 
$792,032,  Amount  due  Mch.  18,  1870. 

Payment, J?m  I    ^245.04,     ^mown<  $240,  3/cA.  18, 1870. 
Int.  on  same,  vO.04.  J     ^54(3  992^  Amount  due  at  maturity. 

Note. — The  annual  interest  and  the  interest  on  the  same  are  com- 
puted to  the  close  of  the  year  in  which  the  first  payment  is  made,  and 
the  interest  on  the  payment  is  computed  to  the  same  date.  The  differ- 
ence hetween  the  amount  of  the  face  of  the  note  and  the  amount  of  the 
payment  is  the  new  principal  for  the  third  year. 

8.  A  note  of  $500,  dated  Jan.  15,  1865,  and  due  in  2 
years,  with  interest  at  10%,  payable  annually,  is  indorsed 


208  COMPLETE    ARITHMETIC. 

as  follows:  May  21,  1866,  $100;  Mch.  9,  1867,  $200.    What 
was  the  amount  due  July  15,  1867  ? 

335.  Rules. — 1.  To  compute  unpaid  annual  interest, 
Compute  the  interest  on  the  iwincipal  for  the  entire  time  it  is 
on  interest,  and  the  interest  on  each  year's  interest  for  the  time 
it  is  unpaid.  The  sum  of  the  principal,  the  interest  on  the 
principal,  and  the  interest  on  Hie  unpaid  interest  will  be  the 
amount  due. 

Note. — Instead  of  computing  the  interest  on  the  several  annual 
interests  separately,  simple  interest  may  be  computed  on  one  year's  inter- 
est for  a  time  equal  to  the  sum  of  the  periods  of  time  the  several  annual 
interests  are  unpaid. 

2.  To  compute  annual  interest  when  partial  payments 
have  been  made,  1.  Compute  the  interest  on  the  principal  to  the 
end  of  the  first  year  in  which  any  payment  is  made,  and  also 
the  interest  on  the  unpaid  annual  interest  to  the  same  date,  and 
f(yrm  the  amount. 

2.  Compute  the  interest  on  the  payment  or  payments  to  the 
end  of  the  year,  and  form  tJie  amount. 

3.  Subtract  the  amount  of  the  payment  or  payments  from  the 
amount  of  the  principal  and  interest,  arid  taking  the  difference 
for  a  new  principal,  proceed  as  before  with  succeeding  pay- 
ments, making  the  date  of  settlement  the  last  date. 

Note. — 1.  This  rule  is  based  on  the  principle  that  the  payments 
with  added  interest  should  be  applied  first  to  the  discharge  of  the 
accrued  interest  at  the  end  of  a  year,  and  then  to  the  discharge  of  the 
principal.  When  the  amount  of  the  payment  or  payments  will  not 
cancel  all  the  interest  due,  the  unpaid  interest  draws  simple  interest  to 
the  end  of  the  next  year  in  which  a  payment  is  made. 


COMPOUND  INTEREST. 

336.  Compound  Interest  is  interest  on  the  princi- 
pal and  also  on  the  interest  which,  at  regular  intervals  of 
time,  is  added  to  the  principal.  It  is  generally  compounded 
annually,  semi-annually,  or  quarterly. 


COMPOUND  INTEREST.  209 


WRITTEN  PROBLEMS. 


1.  A  note  of  S450  is  due  in  3  years  with  interest  at  6  %, 
compounded  annually.  What  will  be  the  amount  due  at 
maturity?    What  will  be  the  compound  interest  due? 


Process. 

$450 

.06 

$27.00 

1st  year's  interest. 

450. 

$477 

'Id  principal. 

.06 

$28.62 

2d  yearns  interest. 

477. 

$505.62 

3d  principal. 

.06 

$30.3372 

3d  year\  interest. 

505.62 

$535.9572 

Amount  due  at  the  end  of  the  third  year. 

450. 

$85.9572 

Compound  interest  at  the  end  of  the  third  year. 

2.  What  is  the  amount  of  $600  for  4  years  at  5  %,  com- 
pounded annually?     What  is  the  compound  interest? 

3.  What  is  the  compound  interest  of  $1500  for  3  years 
at  10  %  ?     At  8  %  ? 

4.  What  is  the  amount  of  $800  for  2  years  at  6  % ,  com- 
pounded semi-annually? 

Suggestion.  -Compute  the  interest  at  3  %. 

5.  What  is  the  compound  interest  of  $650  for  3  yr.  4  mo. 
12  d.  at  6  %  per  annum? 

337.  Rule. — To  compute  compound  interest,  Find  the 
amount  of  tJie  given  principal  for  one  interval  of  time;  then, 
taking  tJiis  amount  as  a  new  principal,  find  the  amount  for  the 
second  interval,  and  so  continue  for  the  entire  time.  The  dif- 
ference between  the  UlsI  amount  and  the  principal  is  the  com- 
pound interest  for  the  time. 

C.Ar.— 18. 


210 


COMPLEl'E  ARITHMETIC. 


Notes. — 1.  When  the  interest  is  compounded  semi-annually,  the 
rate  per  cent,  is  one  half  the  yearly  rate,  and  when  compounded 
quarterly,  it  is  one  fourth  the  yearly  rate. 

2.  When  the  time  contains  years,  months,  and  days,  the  amount 
is  found  for  the  number  of  whole  intervals  in  the  time,  and  then  the 
interest  is  computed  on  this  amount  for  the  remaining  months  and 
days. 

338.  Compound  interest  is  usually  computed  by  the  aid 
of  a  table  giving  the  amount  of  $1  at  several  different  rates 
per  cent.,  and  for  any  number  of  years  which  may  be  in- 
cluded. 

339.  A  Table 

Slioiving  the  amount  of  $1  at  compound  interest,  at  3,  4,  5,  6, 
7,  or  8  per  cent.,  for  any  number  of  years  from  1  to  25. 


YRS. 

3  PER  CENT. 

4  PER  CENT. 

5  PER  CENT. 

6  PER  CENT. 

7  PER  CENT. 

8  PER  CENT. 

1 

1.03 

1.04 

1.05 

1.06 

1.07 

1.08 

2 

1.0609 

1.0816 

1.1025 

1.1236 

1.1449 

1.1664 

3 

1.092727 

1.124864 

1.157625 

1.191016 

1.225043 

1.259712 

4 

1.125509 

1.169859 

1.215506 

1.262477 

1.310796 

1.360489 

5 

1.159274 

1.216653 

1.276282 

1.338226 

1.402552 

1.469328 

6 

1.194052 

1.265319 

1.340096 

1.418519 

1.500730 

,1.586874 

7 

1.229874 

1.315932 

1.407100 

1.503630 

1.605781 

1.718824 

8 

1.266770 

1.368569 

1.477455 

1.593848 

1.718186 

1.850930 

9 

1.304773 

1.423312 

1.551328 

1.689479 

1.838459 

1.999005 

10 

1.343916 

1.480244 

1.628895 

1.790848 

1.967151 

2.158925 

11 

1.384234 

1.539454 

1.710339 

1.898299 

2.104852 

2.331639 

12 

1.425761 

1.601032 

1.795856 

2.012196 

2.252192 

2.518170 

13 

1.468584 

1.665074 

1.885649 

2.132928 

2.409845 

2.719624 

14 

1.512590 

1.731676 

1.979932 

2.260904 

2.578534 

2.937194 

15 

1.557967 

1.800944 

2.078928 

2.396558 

2.759032 

3.172169 

16 

1.604706 

1.872981 

2.182875 

2.540352 

2.952164 

3.425943 

17 

1.652848 

1.947900 

2.292018 

2.692773 

3.158815 

3.700018 

18 

1.702433 

2.025817 

2.406619 

2.854339 

3.379932 

3.996019 

19 

1.753506 

2.106849 

2.526950 

3.025600 

3.616528 

4.315701 

20 

1.806111 

2.191123 

2.653298 

3.207135 

3.869684 

4.660957 

21 

1.860295 

2.278768 

2.785963 

3.399564 

4.140562 

5.033834 

22 

1.916103 

2.369919 

2.925261 

3.603537 

4.430402 

5.436540 

23 

1.973587 

2.464716 

3.071524 

3.819750 

4.740530 

5.871464 

24 

2.032794 

2.563304 

3.225100 

4.048935 

5.072367 

6.341181 

25 

2.093778 

2.665836 

3.386355 

4.291871 

5.427433 

6.848475 

INTEREST.  211 

340.  The  amount  of  $1  for  ike  given  time  and  rate,  multiplied 
by  the  given  j^rincipal,  gives  its  amount  for  the  same  time  and 
rate. 

Note. — When  the  interest  is  compounded  semi-annually,  it  is  com- 
puted from  the  table  by  taking  one  half  the  rate  and  twice  the  number 
of  years. 

6.  What  is  the  compound  interest  of  $750  for  15  years 
at  6  %  ?     What  is  the  amount  ? 

7.  What  is  the  amount  of  $500  for  6  years  at  8  % ,  com- 
pounded semi-annually  ? 

8.  What  is  the  amount  of  $1250  for  10  yr.  4  mo.  15  da. 
at  5  ^ ,  compound  interest  ? 


EQUATION  OF  PAYMENTS. 

341.  Equation  of  I^ayments  is  the  process  of 
finding  an  equitable  time  for  the  payment  of  several  debts, 
due  at  different  times,  without  interest.  It  is  also  called 
the  Average  of  Payments. 

The  equitable  time  sought  is  called  the  Average  Time,  or 
the  Equated  Time. 

PEOBLEMS. 

1.  A  owes  B  $300,  of  whicli  $200  is  due  in  3  months, 
and  $100  in  6  months:  when  will  the  payment  of  $300 
equitably  discharge  the  debt  ? 

Process.  s^  jg  entitled  to  the  use   of   $200    for  3 

$200  X  3  =  $600  months,  which  equals  the  use  of  $600  for  1 

$100  X  6  =  $600  month,  and  to  the  use  of  $100  for  6  months, 

$300  )  $1200  which  equals  the  use  of  $600  for  1  month ; 

4  and,  hence,  he  is  entitled  to  the  use  of  $300 

Ans.   4  mos.        ^^"^il   it   equals  the  use  of  $600  +  $600,  or 

$1200,  for  1  month.    It  will  take  $300  as  many 

months  to  equal  the  use  of  $1200  for  1  month,  as  $300  is    contained 

times  in  $1200,  which  is  4.     Hence,  the  payment  of  $300  in  4  months 

will  equitably  discharge  the  debt. 


212  COMPLETE  ARITHMETIC. 

Proof. — In  paying  the  $200  in  4  months,  A  gains  the  use  of  $200 
for  1  month,  and  in  paying  the  $100  in  4  months,  he  loses  the  use 
of  $100  for  2  months,  which  equals  the  use  of  $200  for  1  month. 
Hence,  his  gain  and  loss  are  equal. 

2.  A  owes  a  merchant  $200  due  in  4  months,  and  $600 
due  in  8  months:  what  is  the  equated  time  for  the  pay- 
ment of  both  debts? 

3.  A  owes  B  $1200,  of  which  $300  are  due  in  4  months, 
$400  in  6  months,  and  the  remainder  in  12  months :  what 
is  the  equated  time  for  the  payment  of  the  whole? 

4.  A  owes  B  $800,  of  which  |  is  due  in  2  months,  ^  in 
3  months,  and  the  remainder  in  6  months :  what  is  the 
equated  time  for  the  payment  of  the  whole? 

5.  A  man  owes  $300  due  in  4  months,  $600  due  in  5 
months,  and  $700  due  in  10  months :  what  is  the  equated 
time  for  the  payment  of  the  whole? 

6.  Smith  &  Jones  bought  $500  worth  of  goods  on  4 
months'  credit,  $700  worth  on  6  months'  credit,  and  $1000 
worth  on  5  months'  credit:  what  is  the  equated  time  for 
the  payment  of  the  whole  ? 

7.  A  bought  $2000  worth  of  goods,  ^  of  which  was  to  be 
paid  down,  i  in  3  months,  ^  in  4  months,  and  the  re- 
mainder in  8  months :  what  is  the  equated  time  for  the 
payment  of  the  whole? 

8.  AVhat  is  the  equated  time  for  the  payment  of  $220, 
due  in  30  days;  $300,  due  in  40  days;  $250,  due  in  60 
days;  and  $100,  due  in  90  days? 

9.  What  is  the  equated  time  for  the  payment  of   $300, 

due  in  30  days;   $250,  due  in  45  days;   and  $350,  due  in 

60  days  ? 

Process  by  Interest. 

Int.  of  $300  for  30  days,  at  6%  =  $1.50 
Int.  of  $250  for  45      "      "  6%  =    1.875 
Int.  of  $350  for  60      "      "  6%=r=    3.50 
$900  $6,875 

$9.00  =  Int.  of  $900  for  60  days. 
.15  =    "     "  $900  for  1  day! 
$6,875  H-  $.15  =  45.9.     Ans,,  46  days. 


EQUATION  OF  PAYMENTS.  213 

The  debtor  is  entitled  to  the  use  (1)  of  $300  for  30  days,  which,  at 
6%,  equals  $1.50  interest;  (2)  of  $250  for  45  days,  which  equals 
$1,875  interest;  (3)  of  $350  for  60  days,  which  equals  $3.50  interest. 
Hence,  he  is  entitled  to  the  use  of  $900,  the  sum  of  the  debts,  until 
the  interest  thereon,  at  6%,  equals  the  sum  of  $1.50  +  $1,875  +  $3.50, 
which  is  $6875.  The  interest  of  $900  for  1  day  is  $.15;  and  since 
$6,875  ^$.15  =  45.9,  it  will  take  45.9  days  for  $900  to  yield  $6,875 
interest.     The  equated  time  for  payment  is  46  days. 

Note. — When  the  fraction  of  a  day  in  the  equated  time  is  more 
than  i,  it  is  counted  as  a  day ;  when  it  is  less  than  ^,  it  is  disre- 
garded. 

10.  What  is  the  equated  time  for  the  payment  of  $520, 
due  in  45  days ;  $340,  due  in  60  days ;  and  $640,  due  in 
90  days  ? 

11.  What  is  the  equated  time  for  the  payment  of  $375, 
due  now  ;  $425,  due  in  30  days  ;  $500,  due  in  60  days ; 
and  $600,  due  in  75  days  ? 

12.  What  is  the  equated  time  for  the  payment  of  $340, 
due  May  10,  1870;  $450,  due  June  10;  $560,  due  July 
15;  and  $650,  due  Aug.  10? 

Note. — Begin  with  the  first  date  (May  10),  and  find  the  exact 
number  of  days  between  it  and  each  succeeding  date.  The  equated 
time  is  counted  forward  from  the  first  date. 

13.  What  is  the  equated  time  for  the  payment  of  $1000, 
due  June  1,  1870;  $850,  due  July  1;  $750,  due  Sept.  1; 
and  $900,  due  Oct.  1  ? 

14.  What  is  the  equated  time  for  the  payment  of  $75, 
due  May  6,  1870;  $115,  due  May  26;  $220,  due  June  25; 
$315,  due  July  16;  and  $350,  due  July  30? 

PRINCIPLES  AND  RULES. 

342.  The  time  between  the  contraction  of  a  debt  and  its 
payment  is  called  the  Term  of  Credit,  or  Time  of  Credit. 

343.  Principles. — 1.  The  payment  of  a  sum  of  money 
BEFORE  it  is  due  is  offset  by  keeping  an  equal  sum  of  money 
an  equal  time  after  it  is  due. 

2.  The  use  of  any  sum  of  money  is  measured  by  its  interest 
for  the  tims. 


214  COMPLETE  ARITHMETIC. 

344.  Rules. — To  equate  the  time  of  several  debts  or 
payments, 

1.  Multiply  each  debt  or  payment  by  its  time  of  credit,  and 
divide  the  sum  of  Hie  products  by  the  sum  of  the  debts  or  pay- 
ments.    Or, 

2.  Compute  the  interest  of  each  debt  or  payment  for  its  time 
of  credit,  and  divide  the  sum  of  the  interests  by  the  interest  of 
the  sum  of  the  debts  or  payments  for  one  month  or  one  day. 

Notes. — 1.  As  the  result  will  be  the  same  at  any  rate,  the  interest 
may  be  computed  at  that  rate  which  is  most  convenient. 

2.  The  correctness  of  each  of  the  above  methods  has  been  called 
in  question  by  a  number  of  authors,  who  commend  the  following  as 
"  the  only  accurate  rule  "  : 

*^Find  the  present  worth  of  each  of  the  given  amounts  due  ;  then  fnd  in 
what  time  the  sum  of  these  present  ivorths  will  amount  to  the  sum  of  all  the 
payments. 

The  inaccuracy  of  this  so-called  "accurate  rule"  is  easily  shown. 
The  raetliods  given  above  are  both  strictly  accurate,  and  they  are  in 
general  use.     (See  appendix.) 

345.  When  partial  payments  are  made  on  a  debt  before 
it  is  due,  the  time  for  the  payment  of  the  balance  of  the 
debt  is  proportionately  extended. 

15.  A  owes  a  merchant  $200,  due  in  12  months,  without 
interest;  in  4  months  he  pays  $50  on  the  debt,  and  in  8 
months,  $50 :  when  in  equity  should  he  pay  the  balance  ? 

Process.  In  paying  $50  in  4  months,  A  loses 

$50X8  =  $400        its  use  for   8   months,  and   in   paying 

$50  X  4  ^  $200        $50  in  8  months,  he  loses  its  use  for  4 

$200  —  $100  =  $100  )  $600        months,  and  hence  he  loses  the  use  of 
6  $400  H-  $200,  or  $600,  for  1  month.     To 

offset  this  loss,  he  is  entitled   to  keep 
the  balance  ($100)  6  months  after  its  maturity. 

16.  A  owes  B  $300,  due  in  8  months  :  if  he  pay  $200 
in  5  months,  when  should  he  pay  the  balance? 

17.  A  man  bought  a  horse,  agreeing  to  pay  $150  in  6 


EQUATION  OF  ACCOUNTS.  215 

months,  without  interest :  if  he  pay  $50  down,  when  should 
he  pay  the  balance? 

18.  A  owes  B  $600,  payable  in  6  months,  but,  at  the  close 
of  3  months,  he  proposes  to  make  a  payment  sufficiently 
large  to  extend  the  time  for  the  payment  of  the  balance  6 
months.     How  large  a  payment  must  he  make  ? 

19.  A  owed  B  $1500,  due  in  12  months,  but  in  4  months 
paid  him  $400,  and  in  6  months  $500 :  when  in  equity  ought 
the  balance  to  be  paid? 

20.  Clark  and  Brown  bought  March  10,  1870,  a  bill  of 
goods  amounting  to  $2500,  on  4  months'  credit;  but  they 
paid  $650  Apr.  7;  $500  Apr.  30;  and  $350  May  20. 
When  ought  they  to  pay  the  balance? 

346.  Rule. — Multiply  each  payment  by  the  time  it  ivas  paid 
before  it  ivas  due,  and  divide  the  sum  of  the  products  by  the  bal- 
ance unpaid. 


EQUATION  OF  ACCOUNTS. 

347.  JEquation  of  Accounts  is  the  process  of  find- 
ing the  equated  time  for  the  payment  of  the  balance  of  an 
account*;  or  the  time  when  the  balance  was  due. 

Case  I. 

7?Lccoian.ts  ooiataining  only  X)eTDit  Items. 
PROBIiEMS. 

1.  A  bookseller  bought  of  Wilson,  Hinkle  &  Co.  the 
following  bills  of  goods,  on  4  months'  credit : 

Feb.      3,  1870,  a  bill  of  $450. 

24,  ''             "          500. 

Mch.   25,  *'            "          750. 

Apr.    20,  '^             "          -600. 

What  is  the  equated  time  of  maturity  ? 


216  COMPLETE  ARITHMETIC. 

Process. 
.      Due  June     3,  1870,  $450  X  00  =^ 

"  24,  "  500X21  =  10500 
"  July  25,  "  750  X  52 --=  39000 
"     Aug.  20,     "         600  X  78  =  46800 

$2300  )  $96300  (  41.8  days. 

The  equated  date  of  maturity  of  the  above  bills  is  42  days  from 
June  3,  1870,  which  is  July  15,  1870. 

Notes. — 1.  The  date  of  maturity  of  each  bill  is  found  by  counting 
'forward  4  months  from  the  date  of  purchase.  The  same  result  would 
be  obtained  by  finding  the  average  or  equated  date  of  purchase,  and 
counting  forward  4  months. 

2.  The  equated  time  of  maturity  may  also  be  found  by  beginning 
at  the  last  date,  and  taking  the  exact  number  of  days  between  each 
preceding  date  and  the  last  date  for  a  multiplier.  The  equated  date 
is  then  found  by  counting  back  from  the  last  date. 

2.  Murray  &  Co.  bought  of  Smith  &  Moore  goods  as 
follows : 

Apr.   15,  1869,  a  bill  of  $400,  on  3  mo.  credit. 
May    20,     ''  ''  245,  on  4    '' 

June  25,     ''  *'  375,  on  4    *' 

Sept.  15,     *'  ''  625,  on  3    " 

What  is  the  equated  time  of  maturity  ? 

3.  A  merchant  has  the  following  charges  against  a  cus- 
tomer : 

May     9,  1870,  $340,  on  4  mo.  credit. 

June     6,     ''         530,  on  4    '' 
July     8,     ''        213,  on  3    ** 
Aug.  30,     "        150,  on  4   " 
What  is  the  equated  time  of  maturity? 

4.  J.  O.  Bates  &  Co.  bought  of  Smith  &  Brown  several 
bills  of  goods,  as  follows : 

March    3,  1868,  a  bill  of  $250,  on  3  mo.  credit. 
April    15,     '*  "  180,  on  4   *' 

June     20,     ''  **  325,  on  3    '' 

Aug.     10,     *'  ''  80,  on  3    '' 

Sept.       1,     "  "  100,  on  4   *' 

What  is  the  equated  date  of  maturity?     How  much  would 

pay  the  account  Dec.  1,  1868? 


EQUATION  OF  ACCOUNTS. 


217 


348.  Rule.— To  find  the  equated  time  of  maturity  for 
the  debit  items  of  an  account,  First  find  the  maturity  of  each 
item  or  hill,  and  then,  counting  from  the  first  date  for  the  time 
of  credit,  find  the  equated  time  as  in  the  equation  of  payments. 
The  date  of  the  equated  time  is  found  by  counting  forivard  from 
the  first  date. 

Notes.— 1.  The  equated  time  may  be  found  by  interest,  as  in  tlie 
Equation  of  Payments.     (Art.  344,  Kule  2.) 

2.  The  sum  of  the  debit  items  draws  interest  from  the  equated  date 
of  maturity  to  the  date  of  payment. 


Case   II. 

A^ccounts   coiataining  Toothi  Debits   a,iad   Credit.-^. 

5.   What  is  the  equated  date  of  maturity  of  each  side  of 
the  following  account? 


IJ7 


John  Smith  in  account  with  John  Jones. 


Cr. 


I.s6^-. 

\TimeofCred. 

1S68. 

Apr.    3, 

To  Mdse. 

$220  I     3  mo. 

July    1, 

By  Cash 

$200 

June    1, 

(( 

125  '     4    " 

Oct.    3, 

u 

150 

July  15, 

u 

200  i     4    " 

Dec.  20, 

« 

300 

Aug.  24, 

li 

140  i     6    " 

i 

Oct.     1, 

« 

190       6     " 

i 

Process. 

Credits. 
Due 

July    1,1868,$200X    00  = 

Oct.     3,     "      150  X    94^=14100 

Dec.  20,     "      300X172^51600 

$650  )  $65700 

101 

Credits  are  due  101  days  from 
July  1,  which  is  Oct.  10. 
Debits  are  due  141  days  from 
July  3,  1868,  which  is  Nov.  21. 

Note. — Each  side  of  the  account  may  be  equated  without  refer- 
ence to  the  other,  as  is  done  above,  or  the  first  or  last  date  of  the  ac- 
count may  be  made  a  common  starting-point  for  both  sides. 
C.Ar.— 19. 


Debits. 

Due 

Julv 

3, 1868,  $220  X 

00- 

= 

Oct. 

1, 

"      125  X 

90  = 

=  11250 

Nov. 

15, 

"       200  X  135  := 

-  27000 

Feb. 

24, 

1869,  140  X  236  = 

=  33040 

Apr. 

1, 

"       190  X 

272- 

^51680 

$875 

)  $122970 

141 

218  COMPLETE  ARITHMETIC. 


6.  The  above  account,  as  equated,  stands  thus: 
Dr. 
Due  Nov.  21,  1868  .    .  S875 


Cr. 

Due  Oct.  10,  1868   .    .  $650 


"When  is  the  balance  of  the  account  due? 

Process. 

I^ebits $875  ^g^Q 

Credits 650  42 

Balance $225  $225  )  $27300 

Difierence  in  time,  42  days.  121 

Balance  is  due  121  days  from  Nov.  21,  1868,  which  is  March  22, 
1869. 

Suppose  the  account  settled  Nov.  21,  the  later  date.  Since  the 
credit  side  of  the  account  has  been  due  since  Oct.  10,  it  has  been  draw- 
ing interest  for  42  days.  To  increase  the  debit  side  of  the  account 
by  an  equal  amount  of  interest,  the  balance  must  remain  unpaid  121 
days.  Counting /orwar<i  121  days  from  Nov.  21,  the  balance  is  found 
to  be  due  March  22,  1869. 

7.  Suppose  that  the  debit  and  credit  sides  of  an  account 
when  equated  stand  as  follows : 

Dr.  I  Or. 

Due  Nov.  21,  1868  .    .  $650  I  Due  Oct.  10,  1868   .    .  $875 

What  would  be  the  equated  time  of  payment  for  the  bal- 
ance ? 

Process. 

Credits $875  _  ^g-c 

Debits    . 650  42 

Balance $225  $225  )  $36750 

Difference  in  time,  42  days.  163 

Balance  is  due  163  days  prior  to  Nov.  21,  1868,  which  is  June  11, 

i868. 

Suppose  the  account  settled  Nov.  21,  as  before.  The  credit  side, 
having  been  due  since  Oct.  10,  has  been  drawing  interest  for  42  days. 
That  the  debit  side  of  the  account  may  be  increased  by  an  equal 
amount  of  interest,  the  balance  must  be  regarded  as  due  163  days 
prior  to  Nov.  21. 


EQUATION  OF  ACCOUNTS. 


219 


8.  The  debit  and  credit  sides  of  an  account  when  equated 

stand  as  follows : 

Br. 


Due  June  5,  1870  .    .  $1285 


Due  July  1,  1870 


Cr. 

$1000 


What  is  the  equated  time  of  payment  for  the  balance  ? 

9.  At  what  time  did  the  balance  of  the  following  equated 
account  begin  to  draw  interest : 

Dr.  I  Or. 

Due  July  12,  1870  .     .  $450  [Due  Sept.  1,  1870    .    .  $800 

10.  When  will  the  balance  of  the  following  account 
begin  to  draw  interest,  the  debit  items  having  a  credit  of  3 
months  ? 

Dr.      R.  Hill  &  Co.,  in  account  with  O.  Cooke.         Cr. 


1S70. 

ISTO. 

July  10 

To  Mdse. 

$120 

Nov.  20 

Bv  Cash 

$350 

"     30 

« 

450 

Dec.  25 

"    Mdse. 

250 

Aug.  30 

<( 

380 

ISTI 

Sept.    9 

(( 

560 

"     30 

(( 

400 

Jan.      1 

"    Cash 

750 

349.  Rule. — To  find  the  equated  time  for  the  payment 
of  the  balance  of  an  account, 

1.  Find  the  equated  time  for  each  side  of  the  account. 

2.  Multiply  the  side  of  the  account  which  falls  due  first  by 
the  numbex  of  days  between  the  dates  of  the  equated  time  of  the 
two  sides,  and  divide  the  product  by  the  balance  of  the  account. 

8.  The  quotient  ivill  be  the  number  of  days  to  the  maturity  of 
the  balance,  to  be  counted  forward  from  the  later  equated  date 
when  the  smaller  side  of  the  account  falls  due  first,  and  back- 
ward when  the  larger  side  falls  due  first. 

Notes. — 1.  When  an  account  is  settled  by  cash,  each  side  of  the 
account  is  increased  by  its  interest  from  maturity  to  the  date  of  set- 
tlement, and  the  difference  between  the  two  sides  thus  increased  by 
interest,  is  called  the  Cash  Balance.  Instead  of  adding  the  accrued 
interest  to  each  side,  the  balance  of  interest  may  be  foimd  and  added 


220  COMPLETE  ARITHMETIC. 

to  or  subtracted  from  the  balance  of  items,  according  as  the  two  bal- 
ances fall  upon  the  same  or  upon  opposite  sides  of  the  account.  Thus, 
in  problem  6  above,  the  balance  of  interest,  which  is  the  interest  of 
$650  for  42  days,  fails  on  the  credit  side,  and  the  balance  of  items  on 
the  debit  side.     The  cash  balance  is  $225  —  $3.90,  which  is  $221.10. 

2.  The  cash  balance  may  be  found  directly,  without  equating  the 
account,  by  finding  the  interest  of  each  item  from  its  maturity  to 
the  date  of  settlement,  and  taking  the  difference  between  the  sums 
of  the  debit  interests  and  credit  interests  for  the  balance  of  interest. 
When  the  balance  of  interest  and  the  balance  of  items  fall  on  the 
same  side,  the  cash  balance  is  their  sum ;  when  they  fall  on  opposite 
sides,  the  cash  balance  is  their  difference. 


SECTION  XV. 
RATIO  AND  PROPORTION. 

KATIO. 

350.  The  relation  between  two  numbers  expressed  by 
their  quotient,  is  called  their  Ratio.  The  ratio  of  6  to  2  is 
6  H-  2,  or  3 ;  and  the  ratio  of  2  to  6  is  2  -f-  6,  or  \. 

MENTAL  EXERCISES. 

1.  What  is  the  ratio  of  8  to  4?     24  to  8?     45  to  15? 

2.  What  is  the  ratio  of  6  to  12  ?    12  to  36?     16  to  64? 

3.  What  is  the  ratio  of  42  to  14?    14  to  42?    12  to  30? 

4.  What  is  the  ratio  of  50  to  15?    15  to  50?    80  to  25? 

5.  What  is  the  ratio  of  36  to  16  ?    60  to  25  ?     70  to  40  ? 

6.  What  is  the  ratio  of  45  to  60  ?     18  to  45  ?     75  to  45  ? 

7.  What  is  the  ratio  of  $33  to  $11  ?     $20  to  $50?     $45 
to  $36?     $50  to  $150? 

8.  What  is  the  ratio  of  16  lb.  to  40  lb.?    28  lb.  to  13  lb.? 

9.  What  is  the  ratio  of  ^^  to  fV?    f\  to  tit?    tt  to  A? 

10.  What  is  the  ratio  of  ^  to  ^?     i  to  -J^?     i  to  i? 

11.  What  is  the  ratio  of  |  to  |?     f  to  |?    f  to  J? 

12.  What  is  the  ratio  of  5  to  ^?     J  to  4?     i  to  2^? 


RATIO.  221 


WRITTEN  EXERCISES. 

351.  The  ratio  of  two  numbers  is  expressed  by  placing  a 
colon  between  them.  The  ratio  of  4  to  10  is  denoted  by 
4  :  10,  and  the  ratio  of  ^  to  f  by  l-  :  f.  The  expression 
4  :  10  is  read  the  ratio  of  4  to  10. 

13.  Express  the  ratio  of  7  to  15.     12  to  35.     35  to  17. 

14.  Express  the  ratio  of  2.5  to  7.5.     3.4  to  .62. 

15.  Express  the  ratio  of  f  to  |.     f  to  b\.     2\  to  f. 

16.  What  is  the  value  of  the  ratio  of  112  to  35? 

Process  :     112  :  35  -^  112  ^  35  =  3i,  Ans. 
What  is  the  value  of 

17.  216  :  81  ?       21.  ^\  :  f  ?  25.  6  qt. :  3  pk.  ? 

18.  129  :  215  ?     22.  150  :  l^f?     26.  5  lb.  12  oz. :  17  lb.  4  oz.? 

19.  14.3:6.5?     23.  12i:30i?     27.  2  ft.  6  in. :  12  ft.  6  in. ? 

20.  1.44 :  3.2  ?     24.  34^ :  5|  ?       28.  15  pk. :  12  bu.  2  pk.  ? 

29.  Reduce  24  :  60  to  its  lowest  terms. 
Process  :     24  :  60  =  |  *  =2^2:5,  Ans. 

Reduce  the  following  ratios  to  their  lowest  terms: 

30.  35  :  84.  33.  105  :  140.  36.   169  :  Qb. 

31.  63  :  108.  34.     81  :  189.  37.  256  :  112. 

32.  121  :  220.  35.   105  :  195.  38.  225  :  120. 

39.  Reduce  f  :  J  to  an  equal  ratio  with  integral  terms. 

Process  :     t  :  f  =  if  :  fV  =^  10  :  9. 

Reduce  the  following  ratios  to  equal  ratios  with  integral 


rms  : 

40.    1  :   3. 

43.    1  :  f 

46. 

*: 

:10. 

41.    f  :  +i. 

44.  ^  :  1^. 

47. 

2i: 

'h 

42.  3^  :  f. 

45.  H  :  H. 

48. 

14; 

:5i. 

49.  Multiply 

10; 

:21 

by  14  :  15. 

Pbocess:{10^21  =  U.     14:15 


Hence,  (10  :  21)  X  (14  :  15)  =  i^  X  H  -=  l>  Ans. 


222  COMPLETE  ARITHMETIC. 

50.  What  is  the  product  of  9  :  10  and  24  :  33? 

51.  What  is  the  product  of  7  :  15,  25  :  14,  and  24  :  35? 

52.  What  is  the  product  of  12  :  25,  15  :  24,  and  16  :  21  ? 

DEFINITIONS,  PRINCIPLES,  AND  RULES. 

352.  Matio  is  the  relation  between  two  numbers  of  tho 
same  kind  expressed  by  their  quotient. 

353.  The  two  numbers  compared  are  called  the  Terms 
of  the  ratio. 

The  first  term  is  the  Antecedent,  and  the  second  term  the 
Consequent.     The  two  terms  form  a  Couplet. 

354.  The  value  of  a  ratio  is  the  quotient  obtained  by 
dividing  the  antecedent  by  the  consequent. 

When  the  antecedent  is  greater  than  the  consequent,  the  value  of 
the  ratio  is  greater  than  1 ;  when  the  antecedent  is  less  than  tlie  con- 
sequent, the  value  is  less  than  1. 

355.  The  ratio  of  two  numbers  is  expressed  by  placing  a 
colon  (:)  between  them;  as,  5:12.  The  colon  is  called 
the  Si^n  of  Ratio. 

Note. — The  sign  of  ratio  is  the  sign  of  division  with  the  hori- 
zontal line  omitted. 

356.  A  ratio  is  also  expressed  in  the  form  of  a  fraction, 
the  antecedent  being  made  the  numerator  and  the  conse- 
quent the  denominator.     Thus,  5  :  12  =  y2. 

Note. — Several  American  authors  divide  the  consequent  by  the 
antecedent,  thus  reversing  the  positions  of  dividend  and  divisor,  as 
indicated  by  the  sign  of  division.  The  great  majority  of  mathe- 
matical writers  divide  the  antecedent  by  the  consequent. 

Ratios  are  either  Simple  or  Compound. 

357.  A  Simple  Hatlo  is  the  ratio  of  two  numbers; 
as  5  :  8,  or  f  :  f. 

Note. — A  simple  ratio,  having  one  or  both  of  its  terms  fractional, 
is  called  by  several  authors  a  Complex  Ratio. 


RATIO.  223 

358.  A  Compound  JRatio  is  the  product  of  two  or 

more  simple  ratios ;  as,  (5  :  6)  X  (f  •'  10). 

It  may  be  expressed  in  three  ways,  as  follows : 

(5  :  6)  X  (8  :  9)  X  (f  :  10);  or  I  X  I  X  i;  or  8  :  9J^ 

359.  An  Inverse  Hatio  is  a  ratio  resulting  from  an 
inversion  of  the  terms  of  a  given  ratio.  Thus,  5  :  7  is  the 
inverse  of  7  :  5.     It  is  also  called  a  Beciproeal  Ratio. 

360.  Principles. — 1.  The  two  terms  of  a  ratio  must  be  like 
numhers. 

2.  Tlie  antecedent  equals  the  consequent  multiplied  by  the 
ratio. 

3.  The  consequent  equals  the  antecedent  divided  by  the  ratio. 

4.  If  the  product  of  the  two  terms  of  a  ratio  be  divided  by 
either  term,  the  quotient  will  be  the  other  term. 

5.  A  ratio  is  multiplied  by  multiplying  the  antecedent  or 
dividing  tJie  consequent  by  a  number  greater  than  1. 

6.  A  ratio  is  divided  by  dividing  the  antecedent  or  multiplying 
the  consequent  by  a  number  greater  than  1. 

7.  A  ratio  is  not  changed,  by  multiplying  or  dividing  both  of 
its  terms  by  the  same  number. 

8.  The  product  of  two  or  more  ratios  equals  the  ratio  of  their 
products. 

361.  Rules. — 1.  To  reduce  a  simple  ratio  to  its  lowest 
terms,  Divide  both  terms  by  their  greatest  common  divisor. 
(Pr.  7.) 

2.  To  reduce  a  simple  ratio  with  fractional  terms  to  one 
witli  integral  terms,  Midtiply  both  terms  by  the  least  common 
multiple  of  the  denominators  of  the  fractions.     (Pr.  7.) 

3.  To  find  the  product  of  two  or  more  simple  ratios, 
MuUiply  the  antecedents  together  for  an  antecedent  and  the  'con- 
sequents together  for  a  consequent.     (Pr.  8.) 

Note. — The  process  may  be  shortened  by  cancellation. 


224  COMPLETE  ARITHMETIC. 

PROPORTION. 

MENTAL    EXERCISES. 

The  ratio  of  12  to  6  is  equal  to  the  ratio  of  14  to  7, 
since  the  value  of  each  ratio  is  2. 

1.  What  two  numbers  have  a  ratio  to  each  other  equal 
to  the  ratio  of  15  to  5  ?     24  to  12? 

2.  What  two  numbers  have  a  ratio  to  each  other  equal 
to  6:  24?     7:21?     11:44? 

3.  What  two  numbers  have  a  ratio  to  each  other  equal 
to  45:  15?     12:60?     72:24? 

4.  To  what  number  has  10  a  ratio  equal  to  the  ratio 
of  30  to  15?     14  to  28? 

5.  To  what  number  has   16  a  ratio   equal    to   11  :33? 

6.  To   what   number   has    12   a    ratio   equal   to   6:30? 
24:16?     20  to  15? 

7.  12  is  to  60  as  5  is  to  what  number? 

8.  13  is  to  39  as  15  is  to  what  number? 

9.  14  is  to  42  as  25  is  to  what  number? 

10.  56  is  to  8  as  63  is  to  what  number? 

362.  The  equality  of  two  ratios  is  expressed  by  placing  a 
double  colon  (::)  between  them.  Thus,  5  :  10  =:  7  :  14  is 
written  5  :  10  : :  7  :  14,  and  is  read  5  is  to  10  as  7  is  to  14. 

11.  Read  8  :  40  : :  12  :  60,  and  show  that  the  two  ratios 
are  equal. 

12.  Read  27  :  9  ::  63  :  21,  and  show  that  the  two  ratios 
are  equal. 

13.  Read  5  :  2J  : :  25  :  12^,  and  show  that  the  two  ratios 
are  equal. 

DEFINITIONS  AND  PRINCIPLES. 

363.  A  I*roj)Ortion  is  an  equality  of  ratios. 

364.  The  first  ratio  of  a  proportion  is  called  the  First 
Couplet,  and  the  second  ratio  the  Second  Couplet. 


SIMPLE  PROPORTION.  225 

365.  The  first  and  third  terms  of  a  proportion  are  the 
Antecedents,  and  the  second  and  fourth  terms,  the  Conse- 
quents. 

Note, — The  antecedents  of  a  proportion  are  the  antecedents  of  its 
ratios,  and  the  consequents  are  the  consequents  of  its  ratios. 

366.  The  first  and  fourth  terms  of  a  proportion  are  the 
Extremes,  and  the  second  and  third  terms,  the  Means. 

The  four  terms  of  a  proportion  are  called  Proportionals,  and  the 
last  is  the  fourth  proportional  to  the  other  three  in  their  order. 

367.  Three  numbers  are  in  proportion  when  the  ratio  of 
the  first  to  the  second  equals  the  ratio  of  the  second  to  the 
third ;  as,  8  :  12  : :  12  :  18.  The  second  number  is  called  a 
mean  proportional. 

368.  Proportions  are  either  Simple  or  Compound. 

A  Shnj^le  Proj^ortion  is  an  equality  of  two  simple 
ratios. 

A  Compound  Proportion  is  an  equality  of  two 
ratios,  one  or  both  of  which  are  compound. 

SIMPLE  PROPORTION. 
Cai^e  I. 

A-Tiy  Term  fo"and,  -wlien.  tlie  otliei'  Tliree  Tenns 
are  given. 

369.  The  proportion  4  :  8  : :  6  :  12  may  be  written  4  :  8 
=  6  :  12,  or  f  =  3^  (Art.  356)  ;  and  multiplying  the  two 
equal  fractions  by  12  and  8,  their  denominators,  we  have 
4  X  12  =  6  X  8.     Hence,  the  following 

Principles. — 1.  The  product  of  the  extremes  of  a  propor- 
tion equals  the  product  of  the  means.     Hence, 

2.  If  the  product  of  the  extremes  of  a  proportion  he  divided 
by  either  mean,  the  quotient  tvill  be  the  other  mean. 

3.  If  the  product  of  the  two  means  of  a  proportion  be  divided 
by  either  extreme,  the  quotient  will  be  the  other  extreme. 


226  COMPLETE  ARITHMETIC. 

-WRITTEN   PROBLEMS. 
Find  the  missing  term  in  the  following  proportions 


14. 

21  : 

7  : :  36  :  -- 

22; 

1  :  1  : :  1  :  — 

15. 

15: 

40  : :  18  :  — 

23. 

*:|::-:| 

16. 

—  : 

24  : :  8  :  32 

24. 

-:2i::i:f 

17. 

—  : 

9 : :  60  :  18 

25. 

i:-::i:i 

18. 

45: 

30  : :  -  :  24 

26. 

$5:  $45::  61b.  :  — 

19. 

2.5 

:  62.5::—:  3.25 

27. 

$.75:  $3  ::  —  :  56 oz. 

20. 

7.2 

:  —  ::  4.7  :  9.4 

28. 

16  men  :  96  men  ::  15  days 

21. 

.25 

:  -    ::  2.5  :  7.5 

29. 

8  horses  :  14  horses  ::  ^  :  - 

370.  Rules. — 1.  To  find  either  extreme  of  a  simple  pro- 
portion, Divide  the  product  of  the  two  means  by  the  other 
extreme. 

2.  To  find  either  mean  of  a  simple  proportion,  Divide  the 
product  of  the  two  extremes  by  the  otJier  mean. 

Case  11. 

The  Solntion   of  T>ro"bleixis    by  Sinaple    I'roportion. 

371.  The  solution  of  a  problem  by  proportion  consists  of 
two  parts,  viz.  : 

1.  The  arranging  of  the  three  given  terms,  called  the 
Statement. 

2.  The  finding  of  the  fourth  term  by  Case  I. 

372.  If  the  required  answer  be  made  the  fourth  term  of 
a  proportion,  the  given  number  of  the  problem,  which  is  of 
the  same  kind  as  the  answer,  will  be  the  third  term,  since 
the  two  terms  of  a  ratio  must  be  like  numbers.     (Art.  360.) 

373.  Of  the  two  remaining  numbers  given  in  the  problem, 
the  greater  will  be  the  second  term  when  the  answer  is  to 
be  greater  than  the  third  term,  and  the  less  will  be  the 
second  term  when  the  answer  is  to  be  less  than  the  third 
term,  otherwise  the  two  ratios  can  not  be  equal. 


SIMPLE  PROPORTION.  227 


'WRITTEN    PROBLEMS. 

30.  If  15  yards  of  cloth  cost  $24,  what  will  40  yards 

cost? 

STATEMENT.  Siiice  the  cost  of  40  yards  is  to  be 

,^     ,      .„     1      ff,.-,,      A  the  answer,  make  $24,  the  cost  of  15 

lo  vd.  :  40  yd.  ::  ^24  :  Ans.  ',  •    ,  .  c 

40  yards,  the  third  term  of  a  proportion ; 

15)  $960  ^"d  since  40  yards  will  cost  more  than 

$64     Arts.        1'^  yards,  the  fourth  term  is  to  be  greater 

than  the  third,  and  hence  the  second 

term  must  be  greater  than  the  first.     Make  40  yards  tlie  second  term 

and  15  yards  the  first,  giving  the  proportion  15  yd.  :  40  yd.  ::  $24  :  cost 

of  40  yard.s,  which,  by  Case  I,  is  found  to  be  $64. 

31.  If  45  sheep  cost  $565,  what  will  140  sheep  cost? 

32.  If  13  tons  of  hay  cost  $97.50,  what  will  1\  tons 
cost  ? 

33.  If  70  acres  of  land  cost  $1875,  what  will  320  acres 
cost? 

34.  If  120  acres  of  land  cost  $3000,  how  many  acres  can 
be  bought  for  $4500  ? 

35.  If  4  lb.  6  oz.  of  butter  cost  $1.75,  what  will  17 J 
pounds  cost? 

36.  If  a  man's  pulse  beat  75  times  in  a  minute,  how 
many  times  will  it  beat  in  8  hours  ? 

37.  If  a  clock  ticks  120  times  in  a  minute,  how  many 
times  does  it  tick  in  9J  hours? 

38.  If  a  comet  move  4°  20'  in  15  hours,  how  far  will  it 
move  in  5  days? 

39.  If  a  garrison  of  160  men  consume  24  barrels  of  flour 
in  6  weeks,  how  many  barrels  will  supply  it  one  year? 

40.  If  24  barrels  of  flour  will  supply  160  men  6  weeks, 
how  many  barrels  will  supply  360  men  the  same  time? 

41.  If  a  vertical  staff  3  feet  high  casts  a  shadow  5  feet 
long,  how  long  a  shadow  will  a  pole  120  feet  high  cast  at 
the  same  time? 

42.  If  a  pole  20  feet  high  casts  a  shadow  12  feet  long, 
how  high  is  the  tree  whose  shadow,  at  the  same  time,  is  90 
feet  long? 


228  COMPLETE  ARITHMETIC. 

43.  If  I  of  a  farm  is  worth  $4500,  what  is  J  of  it  worth? 

44.  If  I  of  a  yard  of  silk  cost  $2.10,  what  will  16i  yards 
cost? 

45.  If  6^  tons  of  hay  cost  $58.75,  how  many  tons  can  be 
bought  for  $173.90? 

46.  At  the  rate  of  5  jDcaches  for  8  apples,  how  many 
apples  can  be  bought  for  5  dozen  peaches? 

47.  If  12  men  can  mow  20  acres  of  grass  in  a  day,  how 
many  acres  can  25  men  mow  ? 

48.  If  9  men  can  build  a  wall  in  15  days,  how  long  will 
it  take  5  men  to  build  it? 

STATEMENT.  The  15  days  is  the  third  term, 

_  f.  ,  -    ■,  1  since  the  answer  is  to  be  in  days. 

5  men  :  9  men  ::  lo  days  :  Ans.         ^^  .        ,      ^  ^  _    ,  ,    . . , 

9  If  it  take  9  men  lo  days  to  build 

5  )  ]^35  a  wall,  it  will  take  5  men  more 

27  days    Ans.         than  15  days,  and  hence  the  an- 
swer,  or    fourth   term,   is  greater 
than  the  third  term,  and  consequently  the  second  term  must  be  greater 
than  the  first  term.     The  proportion  is  5  men  :  9  men  : :  15  days  :  27 
days. 

Note. — The  principle  involyed  in  this  class  of  problems  may  thus 
be  stated  :  The  greater  the  cause,  the  less  the  time  required  to  produce  a 
given  effect;  and,  conversely,  the  greater  the  time,  the  less  the  cattse  required. 

49.  If  a  quantity  of  provisions  will  supply  a  garrison  of 
90  men  125  days,  how  long  will  the  same  provisions  supply 
150  men  ? 

50.  If  15  men  can  harvest  a  field  of  wheat  in  12  days, 
how  many  men  can  harvest  it  in  5  days? 

51.  Divide  90  into  two  parts  whose  ratio  is  equal  to  the 
ratio  of  4  and  5. 

r   (4  +  5)  :  90  : :  4  :  Smaller  part. 
Proportions,  i    , ,   ,   ^      ,,^      .    ^      , 

^   (4  4-  5)  :  90  : :  o  :  Greater  part. 

Note. — These  proportions  are  based  on  the  principle  that  when 
four  numbers  are  in  proportion,  the  sum.  of  the  first  and  second  toyns  is 
to  the  sum  of  the  third  and  fourth  terms  as  the  first  tei-m  is  to  the  ilnrd,  or 
as  the  second  term  is  to  the  fourth. 

52.  Divide  640  into  two  parts  proportional  to  8  and  12. 
To  9  and  11. 


SIMPLE  PROPORTION.  229 

53.  An  estate  worth  $9600  was  divided  between  two  heirs 
in  proportion  to  their  ages,  which  were  15  years  and  17  years 
respectively:  how  much  did  each  receive.'^ 

54.  Two  men,  150  miles  apart,  are  approaching  each 
other,  one  traveling  2  miles  to  the  other  3 :  how  far  will 
each  travel  before  they  meet? 

Jg^^'For  additional  problems  see  Problems  for  Analysis,  p.  239 

PRINCIPLES  AND  RULE. 

374.  Principles. — 1.  The  ratio  of  two  like  causes  equals 
the  ratio  of  their  effects.     Conversely, 

2.  The  ratio  of  two  like,  effects  equals  the  ratio  of  their  causes. 

3.  Tlie  ratio  of  two  like  causes  equals  the  inverse  ratio  of 
their  times.     Conversely, 

4.  TJie  ratio  of  the  times  of  two  like  causes  equals  the  in- 
verse ratio  of  tlie  causes. 

5.  The  two  terms  of  each  couplet  of  a  simple  proportion  must 
be  like  numbers.     (Art.  360,  Pr.  1.) 

6.  The  fourth  term  of  a  proportion  equals  the  product  of  the 
second  and  tliird  terms  divided  by  iJie  first  term.     (Art.  370. ) 

375.  Rule. — To  solve  a  problem  by  simple  proportion, 

1.  Take  for  the  third  term  the  number  which  is  of  the  same 
kind  as  the  answer  sought,  and  make  the  other  two  numbers  the 
first  couplet,  placing  the  greater  for  the  second  term,  when  the 
answer  is  to  be  greater  than  the  third  term;  and  the  LF.Sfi  for 
the  second  term,  when  the  answer  is  to  be  less  than  the  third  term. 

2.  Divide  the  product  of  the  second  and  third  terms  by  the 
first  term,  and  the  quotient  will  be  the  fourth  term,  or  answer. 

Notes. — 1.  When  the  terms  of  the  first  couplet  are  denominate 
numbers,  they  must  be  reduced  to  the  same  denomination. 

2.  The  process  of  finding  the  fourth  term  may  be  shortened  by  can- 
cellation.   The  proportion  15  :  45  ::  27.5  :  —  may  be  completed  thus : 


V  97  fS  ^^•*  ^"^^ 


X$ 


4$ 
27.5 


27.5X3^=82.5 


3.  The  process  of  solving  problems  bv  simple  proportion  is  also 
called  "  The  Rule  of  Three.''' 


230 


COMPLETE   ARITHMETIC. 


COMPOUND  PROPORTION. 
Case  I. 

R.ed.'uctiorx   of   CoinpourLcT.    Ratios    and    lProj>oi'tions 
to  Simp-le   Ones. 


1.  Reduce  the   compound  ratio 


20:80 
4:3 
6:8 


ratio  in  its  lowest  terras. 
Process. 


to  a  simple 


20 
4 
6 


80 

3 

8 


20X4X6:80X3X8 
480  :  1920 
1  :  4,  Ans. 


Or: 
1  :  4,  Am. 


A  compound  ratio 
is  the  product  of  two 
or  more  simple  ra- 
tios (Art.  358),  and 
the  product  of  two  or 
more  simple  ratios  is 
found  by  multiply- 
ing the  antecedents  together  for  an  antecedent,  and  the  consequents 
for  a  consequent  (Art.  361).  Hence,  the  compound  ratio  given  is 
equal  to  20  X  4  X  6  :  80  X  3  X  8,  or  480  :  1920,  which,  by  dividing 
both  terms  by  480,  is  reduced  to  1  :  4. 

The  process  may  be  shortened  by  canceling  the  factors  common  to 
the  product  of  the  antecedents  and  the  product  of  the  consequents. 

Note. — The  process  may  be  explained  directly  by  changing  each 
ratio  to  the  fractional  form,  thus : 

20  :  80 


}- 


1X1  = 


^0  X  ^  X  ^  ^ 


X 


1:4. 


$0  X  ^ 

i  4 

Reduce  the  following  compound  ratios  to  simple  ratios  in 
their  lowest  terms : 


2. 


6 
10 
16 


8 

12 

15 


7:20 
40:21 
12:16 
82  :  45 


6.  Reduce   ]  q  .'  fj  [   :  :  16  :  15  to  a  simple  proportion. 
Su(K*ESTiON. — Reduce  the  compound  ratio  to  a  simple  ratio. 


COMPOUND  PROPORTION. 


231 


Reduce   the   following  compound    proportions    to   simple 


proportions 


I  21  :  10  j    •  •  4^  •  ^^ 

48:336") 

5:8       ,^   : :  12  :  56 
12:5      3 


9.  $192  :  $216 


24:36 
3:4 


10. 


6:9 
12:  3 

15:36 


,„{ 


^5  :  27 
••   M2  :  6 


11. 


tion, 


What  is  the  fourth  term 

5:8 

::13  : 


of  the   compound  propor- 


10 
12 


9 
25 


Process. 


0:tM 
^ 10  :  0  3  : 


Or, 


:  13  :  — 


1 


3::  13 
3 


^10 


$4 
03 

13 


39,  Am. 


3X13=:=  39,  Ans. 


An  inspection  of  the  second  process  shows  that  the  four  numbers 
on  the  right  of  the  vertical  line  are  the  factors  of  the  product  of  the 
means,  and  that  the  three  numbers  on  the  left  are  the  factors  of  the  first 
extreme.  By  canceling  the  factors  common  to  dividend  and  divisor, 
the  fourth  term  is  found  directly. 

Find  the  fourth  term  of  these  compound  proportions; 

20  :  48  )  (5:9 

12.  ^  36  :  15  .^  : :  25  :  —  14.  ^  2.5  :  7.5  ^  • :  6  :  — 

10:4  4:10 


13. 


f  16:35 

J  21  :8 
19:6 
I  12  :  45 


: :  16  :  — 


15. 


21:7 
25  :  10 

4  :  6i  r 
15:12j 


1.. 


::35 


376.  Rules. — 1.  To  reduce  a  compound  ratio  to  a  simple 
ratio,  Multiply  the  antecedents  together  for  an  antecedent,  and 
the  consequents  for  a  consequent. 

2.  To  reduce  a  compound  proportion  to  a  simple  propor- 
tion, Reduce  the  compound  ratio,  or  each  compound  ratio,  if 
there  are  two,  to  a  simple  ratio. 


232 


COMPLETE   ARITHMETIC. 


Case  II. 


Xh.e    Solutioii    of  IProbleiTis    by    Com.pou.nd. 
IProportion. 

16.  If  2  men  can  mow  16  acres  of  grass  in-  10  days, 
working  8  hours  a  day,  how  many  men  can  mow  27  acres 
in  9  days,  working  10  hours  a  day? 

Since  the  answer  required 
Statement.  is  to  be  a  number  of  men. 


16  acres 

9  days 

10  hours 


27  acres 
10  days 
8  hours 


;} 


2  men  :  Ans. 


2  :    3  : :  2  men  :  3  men,  Ans. 


Or. 


^10 

0 

10 


10 

$ 


make  2  men  the  third  term. 
If  the  mowing  of  16  acres 
requires  2  men,  the  mowing 
of  27  acres  will  require  more 
than  2  men,  and  hence  the 
first  ratio  is  16  acres  :  27 
acres,  the  greater  number  be- 
ing the  second  term. 

Tf  10  days  require  2  men, 
9  days  will  require  more  than 
2  men,  and  hence  the  second 
ratio  is  9  days  :  10  days,  the 
3  greater  number  being  the  sec- 

ond term. 
If  working  8  hours  a  day  requires  2  men,  working  10  hours  a  day 
will  require  less  than  2  men,  and  hence  the  third  ratio  is  10  hours  :  8 
hours,  the  less  number  being  the  second  term. 
This  statement  gives  3  men  for  the  fourth  terra. 

Note. — In  determining  which  number  of  each  ratio  of  the  com- 
pound ratio  is  to  be  the  second  term,  reason  from  the  number  in  the 

CONDITION. 

17.  If  12  men  can  build  50  rods  of  wall  in  15  days,  how 
many  men  can  build  80  rods  in  16  days  ? 

18.  If  it  cost  $30  to  make  a  walk  10  feet  wide  and  90 
feet  long,  how  much  will  it  cost  to  make  a  walk  8  feet  wide 
and  225  feet  long? 

19.  If  6  men  can  excavate  576  cubic  feet  of  earth  in  8 
days  of  9  hours  each,  how  much  can  8  men  excavate  in  9 
days  of  10  hours  each  ? 


COMPOUND  PROPORTION.  233 

20.  If  7  horses  eat  35  bushels  of  oats  in  25  days,  how- 
many  bushels  will  15  horses  eat  in  21  days? 

21.  If  a  man  walk  120  miles  in  6  days  of  10  hours  each, 
how  many  miles  will  he  walk  in  16  days  of  8  hours  each  ? 

22.  If  1500  bricks,  each  8  in.  long  and  4  in.  wide,  will 
make  a  walk,  how  many  slabs  of  stone,  each  2  ft.  long  and 

1  ft.  4  in.  wide,  will  be  required  for  the  same  purpose? 

23.  If  the  interest  of  $250  for  9  months  is  $11.25,  what 
is  the  interest  of  $650  for  7  months? 

24.  If  it  cost  $84  to  carpet  a  room  36  ft.  long  and  21  ft. 
wide,  what  will  it  cost  to  carpet  a  room  33  ft.  long  and  27 
ft.  wide? 

25.  If  it  cost  $120  to  build  a  wall  40  ft.  long,  14  ft.  high, 
and  1  ft.  6  in.  thick,  what  will  it  cost  to  build  a  wall  180  ft. 
long,  21  ft.  high,  and  1  ft.  3  in.  thick? 

26.  If  4  men  can  dig  a  ditch  72  rd.  long,  5  ft.  wide,  and 

2  ft.  deep  in  12  days,  how  many  men  can  dig  a  ditch  120  rd. 
long,  6  ft.  wide,  and  1  ft.  6  in.  deep  in  9  days  ? 

27.  If  16  men  can  excavate  a  cellar  50  ft.  long,  36  ft. 
wide,  and  8  ft.  deep  in  10  days  of  8  hours  each,  in  how- 
many  days  of  10  hours  each  can  6  men  excavate  a  cellar 
45  ft.  long,  25  ft.  wide,  and  6  ft.  deep  ? 

28.  If  30  men  can  dig  a  ditch  40  rd.  long,  6  ft.  wide, 
and  3  ft.  deep  in  9  days,  working  8  hours  a  day,  how  many 
men  can  dig  a  ditch  15  rd.  long,  4 J  ft.  wide,  and  2  ft.  deep 
in  12  days,  working  6  hours  a  day? 

Note. — For  additional  problems,  see  Problems  for  Analysis. 

PRINCIPLES   AND   RULE. 

377.  Principles. — 1.  A  compound  proportion,  used  in  the 
solution  of  a  problem,  has  only  one  compound  ratio. 

2.  The  order  of  the  terms  of  each  ratio  composing  the  com- 
pound ratio,  is  determined  as  in  simple  proportion. 

3.  The  fourth  term  of  a  compound  proportion  is  equal  to  the 
product  of  all  the  factors  of  the  second  and  third  terms,  divided 
by  the  product  of  tJie  factors  of  the  first  term. 

r.Ar.— CO 


234  COMPLETE  ARITHMETIC. 

378.  Rule. — 1.  Take  for  the  third  term  the  number  which 
is  of  the  same  kind  as  the  answer  sought,  and  arrange  the  first 
and  second  terms  of  each  ratio  composing  the  compound  ratio  as 
in  simple  proportion. 

2.  Reduce  the  compound  ratio  to  a  simple  ratio,  and  divide 
the  product  of  the  second  and  third  terms  of  the  residting  pro- 
portion by  the  first  term.  The  quotient  will  be  the  fouHh  term, 
or  answer  sought.     Or, 

Divide  the  product  of  all  tJie  factors  of  the  second  and  third 
terms  of  the  compound  proportion  by  the  product  of  the  fa/itors 
of  the  first  term,  shortening  the  process  by  cancellation. 

.  Notes. — 1.  The  terras  of  each  ratio  composing  the  compound  ratio 
are  arranged  precisely  as  they  would  be  if  the  answer  depended 
wholly  on  them  and  the  third  term. 

2.  The  process  of  solving  problems  bv  compound  proportion  is 
also  called  ''The  Double  Rule  of  Three.'' 


PARTNERSHIP. 

379.  A  Partnership  is  an  association  of  two  or  more 
persons  for  the  transaction  of  business. 

A  partnership  is  organized  and  regulated  by  a  contract,  called  arti- 
cles of  agreement.     (Art.  240.) 

380.  A  partnership  association  is  called  a  Company,  Firm, 
or  House,  and  the  persons  associated  together  are  called 
Partners. 

381.  The  money  or  property  invested  in  the  business  by 
the  partners  is  called  Capital,  Joint-stock,  or  Stock  in  Trade. 

When  a  partner  furnishes  capital  but  does  not  assist  in  conducting 
the  business,  he  is  called  a  Silent  Partner. 

382.  Partnership  is  either  Simple  or  Compound. 

In  Shnple  Partnership  the  capital  of  the  several 
partners  is  invested  an  equal  time. 

In  Compound  JPartnership  the  capital  of  the  sev- 
eral partner^  is  invested  an  unequal  time. 


PARTNERSHIP. 


235 


SIMPLE   PARTNERSHIP. 


PROBLEMS. 


1.  A,  B,  and  C  entered  into  partnership  in  business  for 
2  years ;  A  put  in  $3600,  B  $2400,  and  C  $2000,  and  their 
net  profits  were  $3000.     What  was  each  partner's  share  ? 

I.  Process  by  Proportion. 


$3600,  A's  capH.  $8000 
2400,  B's  capH.  $8000 
2000,  C's  cap'l.        $8000 

$8000,  Entire  capital. 


$3600  : :  $3000  :  $1350,  A's  share  of  profits. 

$2400  ::  $3000  :    $900,  B'.s      " 

$2000  : :  $3000  :    $750,  C's      " 

$3000,  Entire  profits. 


Since  the  capital  of  the  several  partners  was  employed  an  equal  time, 
their  shares  of  the  profits  are  proportional  to  their  capitals.  Hence, 
the  entire  capital  is  to  each  partner's  capital  as  the  entire  profits  are 
to  his  share  of  the  profits. 


II.  Process  by  Percentage. 


$3000  -^  $8000  =  .37 h 
$3600  X  -37*  ^  $1350,  A's  share. 
$2400  X  .37*  =  $900,    B's      " 
$2000  X  .37J  =  $750,     C's      " 

Or: 


Since  the  profits  were  equal 
to  .372,  or  37^^  %  of  the  entire 
capital,  each  partner's  share  of 
the  profits  was  equal  to  37J  % 
of  his  capital. 


$3600-^ 

$8000 

-r.45. 

AS 

per 

cent. 

of 

the  cap 

$2400 -f- 

$8000 

=  .30, 

B's 

i( 

(<         ( 

$2000  -4- 

$8000 

=  .25, 

C's 

n 

a           I 

$3000  X  -45  = 

$1350, 

A's 

share  of 

the 

profits. 

$3000  X  .30  := 

$900, 

B's 

a 

<( 

a 

$3000  X 

.25  = 

$750, 

Cs- 

« 

" 

li 

III.  Process  by  Fractional  Parts. 


$3000  -^  $8000  =  t§^^  =  |. 
f  of  $3690  =  $1350,  A's  share. 
f  of  $2400  =  $900,     B's     " 
I  of  $2000  =  $760,     O's      " 


Since  the  profits  were  equal  to 
f  of  the  ^entire  capital,  each  part- 
ner's share  of  the  profits  was  equal 
to  I  of  his  capital. 


236  COMPLETE  ARITHMETIC. 

Or: 

$3600  ^  $8000  =  /^,  A's  part  of  the  capital. 

$2400 -^  $8000  =  T^V,  -^'s     " 

$2000 -^  $8000  =- i,     C's     "         "  "  • 

/^  of  $3000  =  $1350,  A's  share  of  the  profits. 

T%  of  $3000  =$900,     B's     *'         " 

I    of  $3000  =  $750,     C's     "        " 

Note. — Let  the  following  problems  be  solved  by  proportion  and  by 
either  of  the  other  methods,  which  the  teacher  or  pupil  may  prefer. 

2.  A  and  B  were  partners  in  business ;  A  put  in  $5000 
and  B  $4000,  and  their  profits  in  three  years  were  $4500 : 
what  was  each  partner's  share  of  the  profits? 

3.  A,  B,  and  C  formed  a  partnership  in  business ;  A  put 
in  $8000,  B  $4500,  and  C  $3500,  and  their  loss  the  first 
year  was  $3200:  what  was  each  partner's  share? 

4.  A,  B,  and  C  are  partners,  and  B  has  invested  |  as 
much  capital  as  A,  and  C  |  as  much  as  B :  if  their  profits 
amount  to  $6300,  what  will  be  each  partner's  share? 

5.  The  capital  of  two  partners  is  proportional  to  4  and  3; 
their  profits  are  $10000  and  their  expenses  $2300:  what  is 
each  partner's  share  of  the  net  profits? 

6.  A,  B,  and  C  form  a  partnership,  A's  capital  being 
$4000,  B's  $6400,  and  C's  $5600;  they  make  a  net  gain  of 
$3200,  and  then  sell  out  for  $2000:  what  is  each  partner's 
share  of  the  gain  ?     Of  the  proceeds  of  the  sale  ? 

PRINCIPLES  x\ND  RULE. 

383.  Principles. — 1.  The  gain  or  loss  of  a  partnership  is 
shared  by  the  partners  in  proportion  to  the  use  of  the  capital 
invested  by  them,  which  is  its  partnership  value. 

2.  When  the  time  is  equal,  the  use  of  tlie  capital  of  the  several 
partners  is  in  proportion  to  its  amount.     Hence, 

3.  In  a  simple  partnership,  the  gain  or  loss  is  shared  by  the 
partners  in  proportion  to  the  amounts  of  their  capital. 

384.  Rule. — To  divide  the  gain  or  loss  of  a  simple  part- 
nership. Divide  the  gain  or  loss  among  the  several  partners  in 
proportion  to  the  atnounts  of  capital  invested  by  them. 


COMPOUND  PARTNERSHIP.  237 

Notes. — 1.  The  above  principles  and  rule  are  applicable  only  when 
the  several  partners  devote  equal  time  or  render  equal  service  in  car- 
rying on  the  business.  The  division  of  profits  or  losses  is  usually 
settled  by  the  terms  of  the  contract. 

2.  The  problems  in  bankruptcy  (Art.  272)  may  also  be  solved  by 
the  above  methods. 


COMPOUND  PARTNERSHIP. 

7.  A  and  B  formed  a  partnership ;  A  put  in  $3000,  and, 
at  the  close  of  the  first  year,  added  $2000;  B  put  in  $4000, 
and,  at  the  close  of  the  second  year,  took  out  $2000 ;  at  the 
close  of  the  third  year,  the  profits  amounted  to  $3450. 
What  was  each  partner's  share? 

I.  Process  by  Products. 

$3000X1=   »00 
$5000X2  =  $10000 

$13000,  A's  capital  for  1  year. 
$4000  X  2  =    $8000 
$2000  X  1  =  _$2000 

$10000,  B's  capital  for  1  year. 
$13000  +  $10000  =  $23000,  Entire  capital  for  1  year. 
$23000  :  $13000  : :  $3450  :  $1950,  A's  share  of  profits. 
$23000  :  $10000  : :  $3450;  $1500,  B's     " 

Since  A  had  $3000  invested  for  1  year  and  $5000  for  2  years,  the 
use  of  his  capital  was  equivalent  to  the  use  of  $13000  for  1  year. 
Since  B  had  $4000  invested  for  2  years  and  $2000  for  1  year,  the  use 
of  his  capital  was  equivalent  to  the  use  of  $10000  for  1  year.  Hence 
the  profits,  amounting  to  $3450,  should  be  shared  by  them  in  proportion 
to  $13000  and  $10000. 

II.  Process  by  Interest. 

Int.  of  $3000  for  1  yr.  =  $180 
"     "   $5000  for  2  yr.  =  $600 

$780,  Int.  of  A's  capital. 
Int.  of  $4000  for  2  yr.  =  $480 
"     '*   $2000  fori  yr.  =  $120 

$600,  Int.  of  B's  capital. 
$780  +  600  =  $1380,  Int.  of  entire  capital. 
$1380  :  $780  : :  $3450  :  $1950,  A\s  share. 
$1380  :  $600  : :  $3450  :  $1500,  B'a     " 


238  COMPLETE  ARITHMETIC. 

Since  the  use  of  capital  is  represented  by  its  interest  for  the  time, 
the  use  of  A's  capital  is  represented  by  $780,  and  the  use  of  B's  by 
$600.  Hence,  the  profits  ($3450)  should  be  shared  by  them  in  pro- 
portion to  $780  and  $600. 

Note. — The  ratio  of  the  interests  will  be  the  same  whatever  be  the 
rate  per  cent. ;  and  hence  the  interest  may  be  computed  at  any  rate. 

8.  A,  B,  and  C  enter  into  a  partnership  for  4  years, 
A  putting  in  $6000  and  B  $8000.  At  the  close  of  the 
second  year,  A  took  out  $2000  and  B  put  in  $2000  ;  and, 
at  the  close  of  the  fourth  year,  they  divided  $8890  as  net 
profits.     What  was  the  share  of  each? 

9.  A  and  B  entered  into  a  partnership  in  business  for  3 
years,  A's  invested  capital  being  $3500  and  B's  $4500.  At 
the  end  of  the  first  year  they  each  took  out  $1000,  and  B 
was  received  as  a  partner  with  a  capital  of  $2500.  At  the 
end  of  the  third  year  they  dissolved  partnership,  dividing 
$5000  as  net  profits.     What  was  each  partner's  share? 

10.  A,  B,  and  C  entered  into  business  as  partners,  each 
putting  in  $5000  as  capital.  At  the  end  of  2  years  A  took 
out  $1000,  B  $2000,  and  C  $3000,  and,  at  the  end  of  the 
fourth  year,  they  closed  the  business  with  a  loss  of  $3600. 
What  was  the  loss  of  each? 

PRINCIPLE  AND  RULES. 

386.  Principle. — The  value  of  capital  in  compound  partner- 
ship  depends  jointly  on  its  amount  and  the  time  of  its  investment. 

386.  Rules. — To  divide  the  gain  or  loss  of  a  compound 
partnership,  1.  Multiply  the  amount  of  capital  invested  by  each 
partner  by  the  time  of  its  investment,  and  taking  the  product  as 
the  partnership  value  of  his  capital,  proceed  as  in  simple  partner- 
ship.    Or, 

2.  Find  the  interest  of  each  partner's  capital  for  the  time  of  its 

investment,  at  any  rate  per  cent. ;  and  taking  the  interest  thus 
found  as  the  partnership  value  of  his  capital,  proceed  as  in  simple 
partnership. 


PROBLEMS  FOR  ANALYSIS.  239 


PROBLEMS  FOR  ANALYSIS. 

Note. — These  problems  are  here  given  to  afford  an  additional 
drill  in  analysis  and,  if  needed,  in  proportion.  For  the  latter  pur- 
pose, the  teacher  can  select  as  many  problems  as  may  be  necessary. 


MENTAL   PROBLEMS. 

1.  If  7  pounds  of  sugar  cost  91  cents,  what  will  20 
pounds  cost? 

2.  If  12  yards  of  muslin  cost  $1.02,  what  will  20  yards 
cost  ? 

3.  If  -|  of  a  yard  of  silk  cost  $fi,  what  will  f  of  a  yard 
cost? 

4.  If  "I  of  a  barrel  of  flour  cost  $d^,  what  will  -f  of  a 
barrel  cost? 

5.  If  f  of  a  pound  of  coffee  cost  15  cent:*,  what  will  3^ 
pounds  cost? 

6.  A  man  sold  a  watch  for  $120,  which  was  f  of  what 
it  cost  him :  how  much  did  it  cost  ? 

7.  If  40  yards  of  carpeting,  j  of  a  yard  wide,  will  cover 
a  floor,  how  many  yards  of  matting,  IJ  yards  wide,  will 
cover  a  floor  of  equal  size  ? 

8.  Two  men,  traveling  in  the  same  direction,  are  60  miles 
apart;  the  one  in  advance  travels  5  miles  an  hour,  and  the 
other  7  miles  an  hour :  in  how  many  hours  will  the  latter 
overtake  the  former? 

9.  If  a  vertical  staff  3  feet  long  casts  a  shadow  2  feet 
in  length,  how  long  a  shadow  will  a  tree  90  feet  high  cast 
at  the  same  time  of  day? 

10.  If  a  steeple  200  feet  high  casts  a  shadow  150  feet 
long,  what  is  the  height  of  a  pole  which,  at  the  same  time 
of  day,  casts  a  shadow  80  feet  long? 

11.  If  5  men  can  do  a  piece  of  work  in  12  days,  how 
long  will  it  take  6  men  to  do  it? 

12.  If  8  men  can  do  a  piece  of  work  in  15  days,  how 
many  men  can  do  the  same  work  in  10  days? 


240  COMPLETE  ARITHMETIC. 

13.  If  9  men  can  do  a  piece  of  work  in  4f  days,  how 
long  will  it  take  7  men  to  do  it? 

14.  If  3  pipes  will  empty  a  cistern  in  30  minutes,  how 
many  pipes  will  empty  it  in  10  minutes  ? 

15.  If  a  quantity  of  provisions  will  supply  15  men  20 
days,  how  long  will  it  supply  50  men  ? 

16.  If  it  require  12  days  of  10  hours  each  to  do  a  piece 
of  work,  how  many  days  of  8  hours  each  will  be  required 
to  do  the  same  work? 

17.  If  5  men  can  do  f  of  a  piece  of  work  in  a  day,  how 
long  will  it  take  one  man  to  do  the  entire  work? 

18.  If  8  men  can  do  |  of  a  piece  of  work  in  3  days,  how 
long  will  it  take  4  men  to  do  the  entire  work? 

19.  If  20  men  earn  $120  in  4  days,  how  much  will  5 
men  earn  in  8  days? 

20.  If  6  men  can  mow  30  acres  of  grass  in  3  days,  how 
many  acres  will  9  men  mow  in  5  days? 

21.  If  5  horses  eat  40  bushels  of  oats  in  3  weeks,  how 
many  bushels  will  supply  12  horses  10  weeks  ? 

22.  If  8  men  can  dig  a  ditch  40  rods  long  in  6  days,  how 
long  will  it  take  12  men  to  dig  a  ditch  60  rods  long? 

23.  If  the  interest  of  $50  for  9  months  is  $6,  what  would 
be  the  interest  of  $150  for  1  yr.  6  mo.? 

24.  A  school  enrolls  180  pupils,  and  the  number  of  boys 
is  f  of  the  number  of  girls :  how  many  pupils  of  each  sex 
are  enrolled  in  the  school  ? 

25.  A  lady  paid  $130  for  a  watch  and  chain,  and  the  cost 
of  the  watch  was  f  more  than  the  cost  of  the  chain  :  what 
was  the  cost  of  each  ? 

26.  A  tree  120  feet  in  height  was  broken  into  two  parts 
by  falling,  and  |  of  the  shorter  part  equaled  }  of  the 
longer :  what  was  the  length  of  each  part  ? 

27.  A  person  giving  the  time  of  day,  said  that  |  of  the 
time  past  noon  equaled  the  time  to  midnight :  what  was  the 
hour  of  day  ? 

28.  A  person  being  asked  the  time  of  day,  said  that  ^  of 
the  time  past  midnight  equaled  the  time  to  noon :  what  was 
the  hour  of  day  ? 


PROBLEMS  FOR  ANALYSIS.  241 

29.  What  is  the  hour  of  day  when  f  of  the  time  pa?t 
noon  equals  |  of  the  time  to  midnight  ? 

30.  What  is  the  time  of  day  when  |  of  the  time  past 
noon,  multiplied  by  4^,  is  equal  to  the  time  to  midnight  ? 

31.  What  is  the  time  of  day  when  f  of  the  time  to  noon 
is  equal  to  f  of  the  time  past  midnight? 

32.  A  man  being  asked  his  age  said,  10  years  ago  my  age 
was  J  of  my  present  age :  what  was  his  age  ? 

33.  A  son's  age  is  f  of  the  age  of  his  father,  and  the  sum 
of  their  ages  is  80  years  :  what  is  the  age  of  each  ? 

34.  Ten  years  ago  the  age  of  A  was  f  of  the  age  of  B, 
and  10  years  hence  the  age  of  A  will  be  f  of  the  age  of 
B :  what  is  the  age  of  each  ? 

35.  At  the  time  of  marriage  a  wife's  age  was  f  of  the 
age  of  her  husband,  and  10  years  after  marriage  her  age 
was  Y^^  of  the  age  of  her  husband :  how  old  was  each  at  the 
time  of  marriage  ? 

36.  f  of  A's  age  equals  f  of  B's,  and  the  difference  be- 
tween their  ages  is  10  years  :  how  old  is  each  ? 

37.  Twice  the  age  of  A  is  20  years  more  than  the  age  of 
B,  and  10  years  more  than  the  age  of  C,  and  the  sum  of 
their  ages  is  120  years :  what  is  the  age  of  each? 

38.  A  man  bought  a  horse  and  carriage  for  $275,  and  ^ 
of  the  cost  of  the  carriage  -|-  $100  was  equal  to  ^  of  the 
cost  of  the  horse :  what  Avas  the  cost  of  each  ? 

39.  A  man  bought  a  horse,  saddle,  and  bridle  for  $150 ; 
the  cost  of  the  saddle  was  ^  of  the  cost  of  the  horse,  and 
the  cost  of  the  bridle  was  ^  of  the  cost  of  the  saddle :  what 
was  the  cost  of  each  ? 

40.  A  man  and  his  two  sons  earned  $140  in  a  month ; 
the  man  earned  twice  as  much  as  the  elder  son,  and  the 
elder  son  earned  twice  as  much  as  the  younger  :  how  much 
did  each  earn  ? 

41.  Two  men  bought  a  barrel  of  sirup,  one  paying  $20 
and  the  other  $30:  what  part  should  each  have? 

42.  Two  men  hired  a  pasture  for  $40,  and  one  put  in  3 
cows  and  the  other  5  cows  :  how  much  ought  each  to  pay  ? 

C.Ar.— 21. 


242  COMPLETE  ARITHMETIC. 

43.  A  and  B  rented  a  pasture  for  $72;  A  puts  in  40 
sheep  and  B  8  cows :  if  4  sheep  eat  as  much  as  one  cow, 
how  much  ought  each  to  pay? 

44.  Two  men,  A  and  B,  agreed  to  build  a  wall  for  $300; 
A  sent  5  men  for  4  days,  and  B  5  men  for  6  days  :  how 
much  ought  each  to  receive? 

45.  A  and  B  engage  to  plow  a  field  for  $81 ;  A  furnished 
3  teams  for  5  days,  and  B  furnished  4  teams  for  3  days  : 
how  much  should  each  receive  ? 

46.  A  man  can  do  i  of  a  piece  of  work  in  a  day,  and  a 
boy  can  do  -J-  of  it  in  a  day:  in  how  many  days  can  both 
of  them,  working  together,  do  it  ? 

47.  A  and  B  together  can  build  a  wall  in  8  days,  and  A 
can  build  it  alone  in  12  days :  how  long  will  it  take  B  to 
build  it? 

48.  A  can  do  a  piece  of  work  in  6  days,  and  B  in  8 
days  :  if  they  both  work  together  3  days,  how  long  will  it 
take  B  alone  to  complete  the  work  ? 

49.  John  can  saw  a  pile  of  wood  in  6  days,  and,  with  the 
assistance  of  Charles,  he  can  saw  it  in  4  days :  how  long 
will  it  take  Charles  to  saw  it  alone  ? 

50.  A  man  can  do  a  piece  of  work  in  5  days  and  a  boy 
in  8  days  ;  the  man  w^orks  2  days  alone  and  is  then  assisted 
by  the  boy:  how  long  will  it  take  both  to  complete  the 
work  ? 

51.  A  and  B  can  do  a  piece  of  w^ork  in  10  days,  and  A, 
B,  and  C  in  8  days :  how  long  will  it  take  C  alone  to  do 
the  work? 

52.  A  and  B  can  do  |^  of  a  piece  of  work  in  a  day,  and 
A  can  do  twice  as  much  in  a  day  as  B :  how  long  will  it 
take  B  alone  to  do  it? 

53.  A  can  make  a  fence  in  ^  of  a  month,  B  in  I  of  a 
month,  and  C  in  -J-  of  a  month :  in  what  time  can  all  three 
together  build  it? 

54.  A  can  do  a  piece  of  work  in  4  days,  B  in  5  days, 
and  C  in  6  days :  in  what  time  can  they  together  do  it? 

55.  A,  B,  and  C  can  do  a  piece  of  work  in  4  days,  A 


PROBLEMS  FOR  ANALYSIS.  243 

and  C  in  8  days,  and  B  and  C  in  12  days :  how  long  will 
it  take  A  and  B  together  to  do  it? 

dQ.  a  and  B  did  a  piece  of  work,  and  f  of  what  A  did 
equaled  |  of  what  B  did :  if  B  received  $18,  how  much  did 
A  receive? 

57.  A  man  spent  f  of  his  money  and  then  earned  ^  as 
much  as  he  had  spent,  and  then  had  $21  less  than  he  had 
at  first:  how  much  money  did  he  have  at  first? 

58.  At  what  time  between  one  and  two  o'clock  will  the 
hour  and  minute  hands  of  a  watch  be  together? 

59.  At  w^hat  time  between  two  and  three  o'clock  are  the 
hour  and  minute  hands  of  a  watch  together  ?  At  what  time 
between  four  and  five  o'clock? 

60.  A  man  has  2  watches,  and  a  chain  worth  $20 ;  if  he 
put  the  chain  on  the  first  watch  it  will  be  worth  f  as  much 
as  the  second  watch,  but  if  he  put  the  chain  on  the  second 
watch  it  will  be  worth  2J  times  the  first  watch :  what  is  the 
value  of  each  watch? 

WRITTEN  PROBLEMS. 

61.  A  father  bequeathed  $14535  to  two  sons,  giving  the 
younger  ^^  as  much  as  the  elder :  what  was  the  share  of 
each  ? 

62.  An  estate  was  so  divided  between  two  heirs  that  |  of 
the  share  of  the  elder  was  equal  to  f  of  the  share  of  the 
younger,  and  the  diflTerence  between  their  shares  w^as  $362  : 
what  was  the  share  of  each  ? 

63.  An  estate  worth  $27520  was  divided  between  two 
daughters  in  proportion  to  their  ages,  which  were  14  and 
18  years  respectively :  how  much  did  each  receive  ? 

64.  A  man  paid  $8100  for  2  farms,  and  |  of  the  cost  of 
the  larger  farm  was  equal  to  y%  of  the  cost  of  the  smaller : 
what  was  the  cost  of  each  ? 

65.  A  earns  $15.50  as  often  as  B  earns  $12.40,  and  in  a 
certain  time  they  together  earn  $697.50:  how  much  did 
each  earn? 

66.  The  fore  wheels  of  a  carrrage  are  each  9^  feet  in  cir- 


244  COMPLETE  ARITHMETIC. 

cumference,  and  the  hind  wheels  are  each  12^  feet  in  cir- 
cumference :  if  each  fore  wheel  revolve  9500  times  in  going 
a  certain  distance,  how  many  times  will  each  hind  wheel 
revolve  ? 

67.  If  it  take  13200  steps  of  2  ft.  9  in.  each  to  walk  a 
certain  distance,  how  many  steps  of  1  ft.  10  in.  each  will 
it  take  to  walk  the  same  distance  ? 

68.  If  $75  yield  $10.80  interest,  what  principal  will  yield 
$89.28  interest  in  the  same  time? 

69.  If  the  interest  of  $475  is  $118.75,  what  would  be  the 
interest  of  $850  for  the  same  time  and  at  the  same  rate? 

70.  If  the  interest  of  a  certain  principal  for  a  certain 
time  at  5  per  cent,  is  $120.50,  wiiat  would  be  the  interest 
of  the  same  principal  for  the  same  time  at  12  per  cent.  ? 

71.  A  broker  sold  90  shares  of  railroad  stock  and  gained 
$315:  how  much  would  he  have  gained  if  he  had  sold  245 
shares  ? 

72.  If  a  gain  of  15  per  cent,  on  a  certain  investment 
yields  $2347.50,  what  would  a  gain  of  24  per  cent,  on  the 
same  investment  yield  ? 

73.  If  the  commission  for  selling  3050  pounds  of  butter 
at  30  cents  a  pound  is  $45.75,  what  would  be  the  com- 
mission for  selling  7500  pounds  at  35  cents  a  pound? 

74.  If  the  annual  dividend  on  $40325  worth  of  mining 
stock  is  $3226,  what  is  the  dividend  on  $70680  of  the  same 
stock? 

75.  If  6  ranks  of  wood,  each  60  ft.  long  and  6  ft.  high, 
are  worth  $337.50,  w^hat  is  the  value  of  15  ranks  of  wood, 
each  45  ft.  long  and  9  ft.  high? 

76.  If  it  cost  $110  to  dig  a  cellar  40  ft.  long,  27  ft.  wide, 
and  4  ft.  deep,  how  much  will  it  cost  to  dig  a  cellar  36  ft. 
long,  30  ft.  wide,  and  5  ft.  deep? 

77.  If  45  men  can  do  a  piece  of  work  in  15  days,  by 
working  8  hours  a  day,  in  how  many  days  can  30  men, 
working  9  hours  a  day,  do  the  same  work? 

78.  If  5  men  can  cut  45  cords  of  wood  in  6  days,  how 
mauv  cords  can  8  men  cut' in  15  davs? 


PROBLEMS  FOR  ANALYSIS.  245 

79.  If  4  men  dig  a  trench  in  15  days  of  10  hours  each, 
in  how  many  days  of  8  hours  each  can  5  men  perform  the 
same  work? 

80.  A  and  B  are  partners  in  business ;  A's  capital  is 
equal  to  f  of  B's,  and  their  profits  are  $3250 :  what  is  the 
share  of  each? 

81.  A  and  B  are  partners ;  f  of  A's  capital  is  equal  to  f 
of  B's,  and  their  loss  in  business  is  $2150:  what  is  the 
share  of  each? 

82.  A,  B,  and  C  are  partners  in  business ;  A's  capital  is 
twice  B's  and  three  times  C's,  and  their  profits  in  business 
are  $4675 :  what  is  the  share  of  each  ? 

83.  A  and  B,  trading  in  partnership  2  years,  make  a 
profit  of  $5460 ;  during  the  first  year  A  owned  |  of  the 
stock,  and  during  the  second  year  B  owned  J  of  it :  what 
is  each  partner's  share  of  the  profits? 

84.  A  and  B,  trading  in  partnership  2  years,  make  a 
profit  of  $2400  ;  A's  capital  the  first  year  was  2|-  times  B's, 
and  the  second  year  it  was  1|-  times  B's :  what  is  each  part- 
ner's share  of  the  profits  ? 

S5.  A  and  B  traded  in  partnership  3  years;  A's  stock 
the  first  year  was  $5000,  the  second  year  $6000,  and  the 
third  year  $7000 ;  B's  stock  the  first  year  was  $7000,  and 
the  last  two  years  $5000;  their  loss  was  $1750.  What  was 
the  loss  of  each  ? 

86.  A  mechanic  agreed  to  work  80  days  on  the  condition 
that  he  should  receive  $1.75  and  board  for  every  day  that 
he  worked,  and  that  he  should  pay  75  cents  a  day  for  board 
when  he  was  idle ;  his  net  earnings  for  the  time  were  $80 : 
how  many  days  did  he  work? 

87.  A  piece  of  carpeting  containing  135  yards  was  cut 
into  3  carpets,  and  -^  of  the  number  of  yards  in  the  first 
carpet  was  equal  to  ^  of  the  number  of  yards  in  the  second 
carpet,  and  to  f  of  the  number  of  yards  in  the  third  carpet : 
what  was  the  number  of  yards  in  each  carpet? 


246 


COMl'LETE  ARITHMETIC. 


SECTION   XVI. 
INYOLUTIO]^  AJ^D  EYOLUTIO]:^ 


I.  INVOLUTION. 


387.  The  first  power  of  4  is  4 ;  the  second  power  of  4  is 
4x4,  which  is  16 ;  the  third  power  is  4  X  4  X  4,  which  is 
64 ;  the  fourth  power  is  4  X  4  X  4  X  4,  which  is  256  ;  etc. 

1.  What  is  the  second  power  of  5?     Of  6?     8?     10? 

2.  What  is  the  third  power  of  3  ?     Of  4  ?    5  ?    6  ?    10  ? 

3.  What  is  the  fourth  power  of  2  ?     Of  3  ?     4?     10  ? 

4.  What  is  the  second  power  of  1  ?     2?     3?     4?     5? 
6?     7?     8?     9? 

5.  What  is  the  third  power  of  1  ?    2  ?    3  ?    4  ?    5  ?     6  ? 
7?     8?     9? 


6.  What  is  the   second   power   of  ^ 


Of  f  ? 


i? 


1  ? 


G   • 


WRITTEN   PROBLEMS. 


7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 


What 
What 
What 
What 
What 
What 
What 
What 
What 
What 


s  the 
s  the 
s  the 
s  the 
s  the 
s  the 
s  the 
s  the 
s  the 
s  the 


second  power  of  406? 
third  power  of  42? 
fourth  power  of  24? 
fifth  power  of  16? 
second  power  of  6.5? 
third  power  of  .  42  ? 
fifth  power  of  .6? 
third  power  of  Jf . 
second  power  of  ||  ? 
fourth  power  of  j|  ? 


Remark. — The  power  to  which  a  number  is  to  be  raised  may  be 
denoted  by  a  little  figure,  called  an  exponent,  placed  at  the  right  of 
the  upper  part  of  the  figures  expressing  the  number.  Thus,  24^ 
denotes  the  second  power  of  24;  16^  denotes  the  third  power  of  16, 
etc. 


INVOLUTION. 

247 

Raise  the  following 

numbers  to  the 

powers  indicated  by 

e  exponents : 

17.  6232 

22.  .0452 

27.  mr 

18.   105^ 

23.   (if)3 

28.  mr 

19.  34.62 

24.  ar 

29.  .005'' 

20.   .016^ 

25.   (161)2 

30.  2.04» 

21.  1.4^ 

26.   (3i)4 

31.  (I)* 

DEFINITIONS  AND  RULE. 

388.  The  Power  of  a  number  is  the  product  obtained 
by  taking  the  number  one  or  more  times  as  a  factor. 

389.  The  First  Power  of  a  number  is  the  number 

itself. 

390.  The  Second  Poiver  of  a  number  is  the  product 
obtained  by  taking  the  number  twice  as 
a  factor. 


It  is  also  called  the  Square  of  the  number, 
since  the  area  of  a  geometrical  square  is  rep- 
resented by  the  product  obtained  by  taking 
the  number  of  linear  units  in  one  of  its  sides 
twice  as  a  factor. 


3X3^ 


391.  The  Third  Poiver  of  a  number  is  the  product 
obtained  by  taking  the  number  three 
times  as  a  factor. 

It  is  also  called  the  Cube  of  the  number, 
since  the  capacity  of  a  geometrical  cube  is 
represented  by  the  product  obtained  by  tak- 
ing the  number  of  linear  units  in  one  of  its 
edges  three  times  as  a  factor. 


3X 


3  ^  27. 


392.  The  Exponent  of  a  power  is  a  small  figure  i:)laced  at 
the  right  of  the  number,  to  show  how  many  times  it  is  to 
be  taken  as  a  factor.     It  denotes  the  degree  of  the  power. 


248  COMPLETE  ARITHMETIC. 

The  first  power  contains  the  number  once  as  a  factor,  and  the  expo- 
nent is  ^  ;  the  second  power,  or  square,  contains  the  number  twice  as 
a  factor,  and  the  exponent  is  2  ;  tlie  tliird  power,  or  cube,  contains 
the  number  three  times  as  a  factor,  and  the  exponent  is  ^  ;  etc. 

393.  Involution  is  the  process  of  finding  the  powers 
of  numbers. 

394.  Rule. — To  raise  a  number  to  a  given  power,  Mul- 
tiply the  number  by  itself  as  many  times  less  one  as  there  are 
units  in  the  exponent  of  the  given  power.  The  last  product  will 
be  the  required  power. 


395.    ATs^OTHER  METHOD   OF   INVOLUTION. 

29.  What  is  the  square  of  53  ? 

Process. 

53  =  50  +  3,  and  53^  =  (50  +  3)2 

50  +  3  Parts  added. 

50  +  3  502  ^  2500 

50X3  +  32   ==(50  +  3)X3  2(50X3)--    300 

502  +  50  X  3  ^  (50  ^  3)  X  50  32  = 9 

502  +  2  (50  X  3}  +  32  =  (50  +  3)2  =  532  =  2809 

An  inspection  of  the  above  process  will  show  that  the  square  of  53 
is  equal  to  the  square  of  the  5  tens,  plus  twice  the  product  of  the  5 
tens  by  the  3  units,  plus  the  square  of  the  units. 

In  like  manner,  it  may  be  shown  that  the  square  of  any  number, 
composed  of  tens  and  units,  is  equal  to  The  square  of  the  tens,  plus  twice 
the  product  of  the  tens  by  the  units,  plus  the  square  of  the  units. 

30.  What  is  the  square  of  45? 

(402  =1600 

Process:  452=  ^i^^X^^Z^^? 

(  2025,  Ans. 

31.  What  is  the  square  of  67?     Of  75? 

32.  What  is  the  square  of  82  ?     Of  38  ? 

33.  What  is  the  square  of  93?     Of  125? 

Suggestion.— 125  =  120  +  5. 

34.  What  is  the  square  of  115?     Of  124? 


EVOLUTION.  249 

35.  What  is  the  cube  of  53  ? 

The  cube  of  53  =  (50  +  3)^  ^50^  +  3  (502X  3)  +  3  (50  X  3^)  -f  3^ 
as  may  be  shown  by  multiplying  50^  +  2  (50  X  3)  +  3^  by  50  +  3. 

In  like  manner,  it  may  be  shown  that  the  cube  of  any  number, 
composed  of  tens  and  units,  is  equal  to  The  cube  of  the  tens,  plus  three 
times  the  product  of  the  square  of  the  fens  by  the  units,  plus  three  times  the 
product  of  the  tens  by  the  square  of  the  units,  plus  the  cube  of  the  units. 

36.  What  is  the  cube  of  45  ? 

r      403  =.  64000 

3  (402  X  5)  =  24000 

Process:  45^  ^  <  3  (40  X  5^)  =    3000 

53  =      125 

*-  91125,  Am. 

37.  What  is  the  cube  of  23?  Of  32? 

38.  What  is  the  cube  of  24?  Of  43? 

39.  What  is  the  cube  of  33?  Of  54? 

40.  What  is  the  cube  of  51  ?  Of  35  ? 

41.  What  is  the  cube  of  45?  Of  52? 

42.  What  is  the  cube  of  41  ?  52  ? 

43.  What  is  the  cube  of  75?  80? 


II.  EVOLUTION. 

MENTAL    PROBLEMS. 

1.  What  are  the  two  equal  factors  of  16?    Of  25?    49? 

2.  Of  what  number  is  81  the  second  power  or  square? 

3.  What  are  the  three  equal  factors  of  8  ?    Of  27  ?    125  ? 

4.  Of  what  number  is  125  the  third  power  or  cube  ? 

One  of  the  two  equal  factors  of  a  number  is  called  its  second  or 
square  root;  one  of  its  three  equal  factors,  its  third  or  cvhe  root;  one 
of  its  four  equal  factors,  its  fourth  root,  etc. 

5.  What  is  the  square  root  of  25  ? 

6.  What  is  the  cube  root  of  8  ? 

7.  What  is  the  cube  root  of  216? 

8.  What  is  the  fourth  root  of  16? 

9.  What  is  the  square  root  of  1  ? 
36?     49?     64?     81? 


Of  49? 

64? 

81? 

Of  27  ? 

64? 

125? 

512?     1000? 

Of  81? 

256? 

625? 

4?    9? 

16? 

25? 

250  COMPLETE   ARITHMETIC. 

10.  What  is  the  cube  root  of  1  ?     8  ?     27  ?     64  ?     125  ? 
216?    343?    512?     729? 

11.  What  integers  between  1  and  100  are  perfect  squares? 

12.  What  integers  between  1  and  1000  are  perfect  cubes  ? 

13.  Show  that  the  square  root  of  a  perfect   square  ex- 
pressed by  two  figures,  can  not  exceed  9. 

14.  Show  that  the  cube  root  of  a  perfect  cube  expressed 
by  three  figures,  can  not  exceed  9. 

DEFINITIONS. 

396.  The  Root  of  a  number  is  one  of  the  equal  factors 
which  will  produce  it. 

397.  The  First  Hoot  is  the  number  itself. 

398.  The  Second  Moot  is  one  of  the  two  equal  factors 
,of  the  number.     It  is  also  called  the  Square  Root. 

399.  The  Third  Moot  is  one  of  the  three  equal  factors 
of  the  number.     It  is  also  called  the  Cube  Boot. 

A  number  is  the  second  power  of  its  square  root;  the  third  power 
of  its  cube  root ;  the  fourth  power  of  its  fourth  root ;  etc. 

400.  A  JPerfect  Power  is  the  product  of  equal  factors. 
It  has  an  exact  root. 

401.  An  Imperfect  Power  is  a  number  which  is  not 
the  product  of  equal  factors.     Its  root  is  called  a  Surd. 

402.  The  Madical  Sign  is  a  character,  ]       ,  placed 
before  a  number  to  show  that  its  root  is  to  be  taken. 

403.  A   small   figure   placed    above  the  radical    sign   is 
called  the  Index  of  the  root. 

Thus,  1^^  denotes  the  first  root  of  25  ;  1^25  denotes  the 
second  root  of  25  ;  f^25,  the  third  root  of  25,  etc. 


SQUARE  ROOT.  251 

"When  the  square  root  is  indicated,  the  index  is  usually  omitted. 
1^16  and  Vl6  alike  denote  the  square  root  of  16. 

Note. — The  root  of  a  number  may  also  be  indicated  by  a  frac- 
tional exponent.  Thus,  16-  denotes  the  square  root  of  16  ;  16*,  the 
cube  root  of  16,  etc. ;  16^  denotes  the  cube  root  of  the  square  of  16. 

404.  EvoltttiOfl  is  the  process  of  finding  the  roots  of 
numbers. 

Note. — Evolution  is  the  reverse  of  involution. 


SQUARE  ROOT. 

Tlie  Division   of   th.e   ]>f\xm'ber'  into   [Periods. 

405.  The  smallest  integer  composed  of  one  order  of  fig- 
ures is  1,  and  the  greatest  is  9 ;  the  smallest  integer  com- 
posed of  two  orders  is  10,  and  the  greatest  is  99,  and  so  on. 

The  squares  of  the  smallest  and  greatest  integers  com- 
posed of  one,  two,  three,  and  four  orders,  are  as  follows  : 

12=              1  92=.              81 

102=         100  992=         9801 

1002  =      10000  9992  =:     997001 

10002  =  1000000  99992  ^  99880001 

A  comparison  of  the  above  numbers  with  their  squares 
shows  that  the  square  of  a  number  contains  twice  as  many 
orders  as  the  number,  or  twice  as  many  orders  less  one. 

406.  Hence,  if  a  number  be  separated  into  periods  of 
two  orders  each,  beginning  at  the  right,  there  will  he  a§  mamj 
orders  in  its  square  root  as  there  are  periods  in  the  numher. 

1.  How  many  orders  in  the  square  root  of  2809? 

Suggestion.— First  divide  the  number  into  periods  of  two  orders 
each,  thus:  2809. 

2.  How  many  orders  in  the  square  root  of  36864? 

3.  How  many  orders  in  the  square  root  of  345744? 


252  COMPLETE  ARITHMETIC. 

4.  How  many  orders  in  the  square  root  of  87616? 

5.  How  many  orders  in  the  square  root  of  5308416  ? 

6.  How  many  orders  in  the  square  root  of  5475600? 

7.  How  many  orders  in  the  square  root  of  14440000? 

407.  The  squares  of  the  smallest  and  greatest  number  of 
units,  tens,  hundredths,  and  thousandths,  are  as  follows  : 

V=              1  92=              81 

102^          100  902=         8100 

1002  ^      10000  9002  ^      810000 

10002  =  1000000  90002  :=  81000000 

A  comparison  of  the  above  numbers  with  their  squares 
shows  that  the  square  of  units  gives  no  order  higher  than 
tens;  that  the  square  of  tens  gives  no  order  lower  than 
hundreds,  nor  higher  than  thousands  ;  that  the  square  of 
hundreds  gives  no  order  lower  than  ten-thousands,  nor 
higher  than  hundred-thousrinds,  etc. 

408.  Hence,  if  a  number  be  separated  into  periods  of  two 
orders  each,  the  left-hand  period  will  contain  the  square  of  the 
left-hand  or  first  term  of  the  square  root ;  the  first  two  left-lmnd 
periods  will  contain  the  square  of  the  first  two  terms  of  the  square 
root,  etc. 

8.  What  is  the  tens'  term  of  the  square  root  of  2025  ? 

Ans. — The  left-hand  period  of  2025  is  20  ;  the  greatest  square  in 
20  is  16,  and  the  square  root  of  16  is  4.  Hence,  the  tens'  figure  of 
the  square  root  of  2025  is  4. 

9.  What  is  the  hundreds'  term  of  the  square  root  of 
87616?     Of  345741? 

10.  What  is  the  left-hand  term  of  the  square  root  of 
16129?     Of  336400? 

11.  What  is  the  left-hand  term  of  the  square  root  of 
87616? 

12.  What  are  the  first  two  terms  of  the  square  root  of 
16129? 


SQUARE  ROOT.  253 


"WRITTEN  PKOBLEMS. 

13.  What  is  the  square  root  of  3364? 

Process.  Since  3364  is  composed  of  two  periods, 

SSfi4  I  58  ^'^  square  root  will  be  composed  of  two 

52  -—  25      orders.     (Art.  406.) 

5  \/  2  =  10  )~864  The  left-hand  period  33  contains    the 

108  X  8  :=    864  square  of  the  tens'  term  of  the  root.     (Art. 

■  408.)     The  greatest  square  in  33  is  25,  and 

the  square  of  25  is  5.     Hence,  5  is  the  tens'  term  of  the  root. 

The  square  of  a  number  composed  of  tens  and  units  is  equal  to 
the  square  of  the  tens  plus  twice  the  product  of  the  tens  by  the  units, 
plus  the  square  of  the  units.  (Art,  395.)  Hence,  the  difference  be- 
tween 3364  and  the  square  of  the  5  tens  of  its  root,  is  composed  of 
tivice  the  product  of  the  tens  of  the  root  by  the  units,  phis  the  square  of  the 
units. 

But  the  product  of  tens  by  units  contain  no  order  lower  than  tens, 
and  hence  the  86  tens  in  the  864,  the  difierence,  contains  ticice  the 
product  of  the  tens  by  the  units.  Hence,  if  the  86  tens  be  divided  by 
twice  the  5  tens  of  the  root,  the  quotient,  which  is  8,  will  be  the  units' 
term  of  the  root. 

If  the  8  units  be  annexed  to  the  10  tens,  used  as  a  trial  divisor,  and 
the  result,  108,  be  multiplied  by  8,  the  product  will  be  twice  the  prod- 
uct of  the  tens  of  the  root  by  the  units,  plus  the  square  of  the  units. 
108X8  =  2  (5X8) +  82. 

Proof.—  58  X  58  =  3364. 

14.  What  is  the  square  root  of  625  ?     Of  4225  ? 

15.  What  is  the  square  root  of  576?     Of  7744? 

16.  What  is  the  square  root  of  1444  ?     Of  6241  ? 

17.  What  is  the  square  root  of  3025?     Of  7569  ? 

18.  What  is  the  square  root  of 

133225  ?  Process. 

19.  What  is  the  square  root  of  133225  j  365 
210681?  -9_    — 

20.  What  is  the  square  root  of  ^  ^.^  ^^  ^  of 
419904?  66X6  =  396_ 
^^^^^''-           .  36X2  =  72)    3625 

21.  What  is  the  square  root  of  725X5—    3^25 
94249?    Of  492804?  : 


254  COMPLETE  ARITHMETIC. 

22.  What  is  the  square  root  of  57600?    Of  40960000? 

23.   What  is  tlie  square  root  of 
Process.  10.4976? 

10.4{)7t)  I  3.24  24.   What  is  the  square  root  of 

1—  176.89? 

^^olTo-^'tA  25.   What  is  the  square  root  of. 

6.2X2— _1.24_  062'!^ 

3.2X2  =  6.4)  .257  6  .oozo  . 

6.44  X  -04  =  .2576  2^-   What  is  the  square  root  of 

^~^  .451584?     Of  .008836? 

27.  What  is  the  square  root  of  586.7? 

Suggestion. — Point  thus  5^6.70,  and  carry  the  root  to  three  deci- 
mal places  by  annexing  periods  of  decimal  ciphers. 

28.  What  is  the  square  root  of  75.364?     Of  5.493? 

29.  What  is  the  square  root  of  263.85?     Of  13467? 

30.  What  is  the  square  root  of  |ff  ?     Of  -^^-fy? 

31.  What  is  the  square  root  of  272^  ?     Of  1040^^ 

32.  What  is  the  square  root  of  2  ?     Of  3  ?     Of  5 1 


_9 

9 


PRINXIFLES  AND  RULE. 

409.  Principles. — 1.  The  square  root  of  a  number  contains 
as  many  orders  as  there  are  periods  of  two  orders  each  in  the 
number. 

2.  The  left-hand  period  of  a  number  contains  the  square  of 
the  first  term  of  its  square  root. 

3.  The  square  of  a  number,  composed  of  tens  and  units,  is 
equal  to  the  square  of  the  tens,  plus  twice  the  product  of  the  tens 
by  the  units,  plus  the  square  of  the  units. 

410.  Rule. — To  extract  the  square  root  of  a  number, 

1.  Begin  at  the  units^  order  and  separate  the  number  into 
periods  of  two  orders  each. 

2.  Find  the  greatest  perfect  square  in  the  left-hand  period, 
and  place  its  square  root  at  the  right  for  the  first  or  highest  term 
of  tJw  root. 

3.  SubtmLct  the  square  of  the  term  of  the  root  found  from  the 


SQUARE  ROOT.  255 

left-hand  'period,  and  to  the  difference  annex  the  second  period 
for  a  dividend. 

4.  Take  twice  the  term  of  the  root  found  for  a  trial  divisor, 
and  the  dividend,  exclusive  of  its  right-hand  figure,  for  a  trial 
diviclend.  The  quotient  {or  the  quotient  reduced)  will  he  the 
next  term  of  the  root. 

5.  Annex  the  second  term  of  the  root  to  the  trial  divisor,  and 
multiply  the  residt  by  the  second  term,  and  subtract  the  product 
from  the  dividend. 

6.  Annex  the  third  period  to  tJie  remainder  for  the  next  divi- 
dend, and  divide  the  same,  exclusive  of  tlie  right-hand  figure,  by 
twice  the  terms  of  the  root  found;  and  continue  in  like  manner 
until  all  the  periods  are  used. 

Notes. — 1.  The  left-hand  period  may  contain  but  one  order. 

2.  Twice  the  term  or  terms  of  the  root,  as  the  case  may  be,  is  called 
a  trial  divisor,  since  the  next  term  of  the  root  is  obtained  from  the 
quotient.  The  term  of  the  root  sought  is  sometimes  less  than  the 
quotient,  since  the  dividend  may  contain  a  part  of  the  square  of  the 
next  term  of  the  root.  The  true  divisor  is  the  trial  divisor  with  the 
next  term  of  the  root  annexed. 

3.  If  the  number  is  not  a  perfect  square,  the  exact  root  can  not  be 
found.  The  exact  root  may  be  approximated  by  annexing  periods 
of  decimal  ciphers.  Since  the  square  of  no  one  of  the  nine  digits 
ends  with  a  cipher,  the  operation  may  be  continued  indefinitely. 

4.  In  pointing  off  a  decimal,  or  a  mixed  decimal  number,  begin 
with  the  order  of  units.  If  there  be  an  odd  number  of  decimal 
places,  annex  a  decimal  cipher. 

5.  When  both  terms  of  a  common  fraction  are  not  perfect  squares, 
the  exact  square  root  can  not  be  found.  An  approximate  root  may 
be  obtained  by  multiplying  both  terms  of  the  fraction  by  tlie  denom- 
inator, and  extracting  the  root  of  the  resulting  fraction.     Thus, 


A'l?.  =  V-7?^  =  ]7,»  "early. 


40       ^     402         40' 
6.  The  square  root  of  a  perfect  square  may  be  found  by  resolving 
it  into  its  prime  factors,  and  taking  the  product  of  one  of  every  two 
of  those  that  are  equal. 

GJ-eonietrical    Explanation. 

411.  The  area  of  a  square  surface  is  found  by  squaring 
the  length  of  one  side ;  and,  conversely,  the  length  of  the 
side  is  found  by  extracting  the  square  root  of  the  number 
(k  noting  the  area.  «* 


256 


COMPLETE  ARITHMETIC. 


Let  the  annexed  diagram  represent  a  square  surface  whose 
area  is  625.  Required  the  length  of  one 
side. 

Since  the  number  denoting  the  area 
contains  two  periods,  there  are  two 
terms  in  the  square  root;  and  since  the 
greatest  square  in  the  left-hand  period 
is  4,  the  tens'  term  of  the  root  is  2. 
(Art.  409.)  Hence  the  length  of  the 
side  of  the  square  is  20  plus  the  units'  term  of  the  root. 
What  is  the  units'  term?  ^ 

Taking  from  the  given  surface  a  square  whose  ¥iHe  is  20 
and  whose  area  is  400,  there  remains  a 
surface  whose  area  is  625  — 400,  or  225. 
This  surface  consists  of  two  equal  rect- 
angles, each  20  in  length,  and  a  small 
square,  the  length  of  whose  side  equals 
the  width  of  each  rectangle.  What  is 
the  width  of  each  rectangle  ? 

Since  the  two  rectangles  contain  most 
of  the  surface  whose  area  is  225,  their  width  may  be  found 
by  dividing  225  by  their  joint  length,  which  is  twice  20,  or 
40.  The  quotient  is  5,  and  hence  the  width  of  each  rect- 
angle is  5,  and  their  joint  area  is  40  X  5,  or  200. 

Removing  the  two   rectangles,   there  remains   the   small 

square,  whose  side  is  5  and  whose  area 

I  ^m     is  25,  the  diiference  between   225  and 

200.  Hence,  5  is  the  units'  term  of 
the  root,  and  the  length  of  the  side  of 
the  square  is  20  -f  5,  or  25. 

Adding  the  area  of  the  several  parts, 

we  have  202  _|_  20  x  5  X  2  +  5^  =  400 

^—~     +  200  4-  25  =r  625. 

It  is  seen  that  the  square  whose  side  is  20,  represents  the 

square  of  the  tens  of  the  root;   the  two  rectangles,  twice  the 

product  of  the  tens  by  the  units ;  and  the  smaller  square,  tlie 

square  of  the  units. 


sax  5 

ji 

^m 

^m 

s^^ 

i^ 

1 

CUBE  ROOT. 257 

Note.— The  entire  length  of  the  surface  whose  area  is  225,  is  twice 
the  side  of  the  square  removed,  plus  the  side  of  the  smaller  square 
(20  X  2  +  5  =  45j,  and  this  multiplied  by  5  gives  an  area  of  225. 


cu:be  root. 

Th-e    Division,    of    the    ISTviiTiber    into    I*ex*iods. 

412.  The  cubes  of  the  smallest,  greatest,  and  an  interme- 
diate number,  composed  of  one,  two,  and  three  orders,  are 
as  follows  : 

V  =  1  93  r=  729  43  =  64 

103  :=        1000        993  =        970299         44^  =        85184 

1003  ==  1000000      9993  =  997002999       444^  =  87528384 

A  comparison  of  the  above  numbers  with  their  cubes 
shows  that  the  cube  of  ar  number  contains  three  times  as 
many  orders  as  the  number,  or  three  times  as  many  orders 
less  two  or  less  one. 

413.  Hence,  if  a  number  be  separated  into  periods  of  three 
orders  each,  there  will  be  as  many  orders  in  its  cube  root  as 
there  are  periods  in  the  number. 

1.  How  many  orders  in  the  cube  root  of  91125? 
Suggestion. — First   point  off  the  number  into  periods  of  three 

orders  each;  thus,  91125. 

2.  How  many  orders  in  the  cube  root  of  84604519? 

3.  How  many  orders  in  the  cube  root  of  912673? 

4.  How  many  orders  in  the  cube  root  of  48228544? 

5.  How  many  orders  in  the  cube  root  of  2357947691  ? 

414.  The  cubes  of  the  smallest  and  greatest  number  of 
units,  tens,  and  hundreds  are  as  follows : 

V  =  1  93  =  729 

103  ==        1000  903  ^        729000 

1003  ^  1000000  9003  =  729000000 

A   comparison   of  the  above  numbers  with  their  cubes 
shows  that  the  cube  of  units  gives  no  order  higher  than 
c.Ar.— 22 


258  COMPLETE  ARITHMETIC. 

hundreds  ;  that  the  cuhe  of  tens  gives  no  order  lower  than 
thousands  nor  higher  than  hundred-thousands;  and  that 
the  cube  of  hundreds  gives  no  order  lower  than  millions 
nor  higher  than  hundred-millions. 

Hence,  if  a  number  be  separated  into  periods  of  three 
orders  each,  tlw  left-haiid  period  will  contain  the  cube  of  the 
first  term  of  the  cube  root;  the  first  two  left-hand  periods  will 
contain  the  cube  of  the  first  two  terms  of  the  cube  root,  etc. 

6.  What  is  the  tens'  terra  of  the  cube  root  of  91125? 

7.  What  is  the  tens'  term  of  the  cube  root  of  912673? 

8.  What   is   the   hundreds'    term   of  the   cube   root  of 
48228544? 

9.  AVhat  is  the  first  term  of  the  cube  root  of  529475129? 

10.  What  is  the  first  term  of  the  cube  root  of  257259456? 

WRITTEN-  PROBLEMS. 

11.  What  is  the  cube  root  of  262144? 

Process.    ^  gj^^^  262144  is  composed  of  two  pe- 

,  262144  [64        riods,  its  cube  root  will  be  composed  of 

6'  =216  4        two    orders    (Art.    413).     The    left-hand 

62  X  3  =  108  )  461 44  period,  262,  contains  the  cube  of  the  tens' 

64 '^       =        262144  term  of  the  root  (Art.  414).     The  greatest 

cube  in  262  is  216,  the  cube  root  of  which 
is  6 ;  hence,  6  is  the  tens'  term  of  the  root.  How  is  the  units'  term 
to  be  found? 

The  cube  of  a  number,  composed  of  tens  and  units,  is  equal  to  the 
cube  of  the  tens,  plus  three  times  the  product  of  the  square  of  the 
tens  by  the  units,  plus  three  times  the  product  of  the  tens  by  the 
square  of  the  units,  plus  the  cube  of  the  units  (Art.  395).  Hence, 
the  difference  between  262144  and  the  cube  of  the  6  tens  of  its  cube 
root,  is  composed  of  three  times  the  product  of  the  square  of  the  tens  of  its 
root  by  the  units,  plus  three  times  the  product  of  the  tens  by  the  square  of  the 
units,  plus  the  cube  of  the  units. 

But  since  the  square  of  tens  gives  no  order  lower  than  hundreds 
(Art.  407),  the  461  hundreds  of  the  difference  (46144)  contains  three 
times  the  product  of  the  square  of  the  tens  by  the  units.  Hence,  if  the  461 
hundreds  (rejecting  the  two  right-hand  figures)  be  divided  by  three 


CUBE  ROOT.  259 

times  the  square  of  the  6  tens  of  the  root,  the  quotient,  which  is  4, 
will  be  the  units'  term  of  the  root.  Cube  64,  and  subtract  the  result  from 
262144.    There  is  no  remainder,  and  hence  64  is  the  cube  root  sought. 

Note. — Instead  of  cubing  64,  the  parts  which  compose  the  differ- 
ence, 46144,  may  be  formed  and  added,  thus: 

^  602X4X3  =  43200 

60  X  42  X  3  =   2880 

43= 64 

46144 

11.  What  is  the  cube  root  of  42875?     Of  91125? 

12.  AVhat  is  the  cube  root  of  117649?     Of  185193? 

13.  What  Is  the  cube  root  of  274625?     Of  405224? 

14.  What  is  the  cube  root  of  704969?     Of  912673? 

15.  What  is  the  cube  root  of  48228544? 

Process. 
48228544  j  364,  Cube  root. 


'6^  =27 
32  X  3  --  27  )  212 
363  ^  46656 


74,  Trial  quotients. 


362X3  =  3888)   15725  _ 
364 '^  =  48228544 

Since  the  two  right-hand  figures  of  each  dividend  are  rejected, 
only  the  first  figure  of  each  period  need  be  brought  down  and  an- 
nexed to  the  difference. 

The  quotient  obtained  by  dividing  212  by  27  is  7,  which  is  too 
large  for  the  second  term  of  the  root,  since  the  cube  of  37  is  more 
than  48228,  the  first  two  periods. 

The  second  difference  is  found  by  subtracting  the  cube  of  36,  the 
first  two  terms  of  the  root,  from  48228,  the  first  two  periods  of  the 
number. 

16.  What  Is  the  cube  root  of  3048625  ?     Of  34328125  ? 

17.  What  is  the  cube  root  of  41063625  ?     Of  43614208  ? 

18.  What  is  the  cube  root  of  27270901  ?    Of  515849608  ? 

19.  What  is  the  cube  root  of  185193?     128024064? 

20.  What  is  the  cube  root  of  103823?     Of  27054036008? 

21.  What  is  the  cube  root  of  15.625  ?     Of  .074256? 

22.  What  is  the  cube  root  of  97.336  ?    Of  .015625  ? 


260  COMPLETE  ARITHMETIC. 

23.  What  is  the  cube  root  of  56.47  ?     Of  12.3456? 

Suggestion. — Point  from  units'  order,  and  fill  decimal  periods, 
thus:  0(3.470,  and  12.345G06. 

24.  What  is  the  cube  root  of  .000042875  ?    Of  67.917312  ? 

25.  What  is  the  cube  root  of  9  ?     Of  31  ?     Of  50  ? 

Suggestion.  — Annex  periods  of  decimal  ciphers  and  carry  the 
root  to  three  decimal  places. 

26.  What  is  the  cube  root  of  2  ?     Of  20  ?     Of  200  ? 

27.  What  is  the  cube  root  of  yVA"  '^     ^^  Hrh  ? 

28.  What  is  the  cube  root  of  llff  ?     Of  37^^? 

29.  A  cubical  box  contains  19683  cubic  inches  :  what  is 
the  length  of  its  edge  ? 

30.  A  block  of  granite  in  the  form  of  a  cube,  contains 
41063.625  cubic  inches  :   what  is  the  length  of  its  edge? 

31.  A  cubical  bin  holds  100  bushels  :  what  is  the  length 
of  its  edge  ? 

32.  If  6  ranks  of  wood,  each  128  ft.  long,  3  ft.  wide,  and 
6  ft.  high,  were  piled  together  in  the  form  of  a  cube,  what 
would  be  the  height  of  the  pile  ? 

PRINCIPLES  AND  RULE. 

415.  Principles. — 1.  The  cube  root  of  a  number  contains 
as  many  orders  as  there  are  periods  of  three  figures  each  in  the 
number. 

2.  The  left-hand  period  of  a  number  contains  the  cube  of  the 
first  term  of  its  cube  root;  the  tivo  left-hand  periods  contain  the 
cube  of  the  first  two  terms  of  the  cube  root,  etc. 

3.  The  cube  of  a  number,  composed  of  tens  and  units,  is 
equal  to  the  cube  of  the  tens,  plus  three  times  the  product  of  the 
square  of  the  tens  by  the  units,  plus  three  times  the  "product  of 
the  tens  by  the  square  of  the  units,  plus  the  cube  of  the  units. 

416.  Rule. — To  extract  the  cube  root  of  a  number, 

1.  Begin  at  the  units'  order  and  separate  tJie  number  into 
periods  of  three  orders  each. 

2.  Find  the  greatest  cube  in  the  left-hand  period,  and  place 
its  cube  root  at  tlie  right  for  Uie  first  term  of  tlie  root. 


CUBE  ROOT.  261 

3.  Subtract  the  cube  of  the  first  term  of  the  root  from  the  left- 
hand  period,  and  to  the  difference  annex  tJie  first  figure  of  the 
next  period  for  a  dividend. 

4.  Take  three  times  the  square  of  the  first  term  of  the  root  for 
a  trial  divisor,  and  the  quotient  for  the  second  term  of  the  root. 
Cube  the  root  now  found,  and,  if  the  residt  is  not  greater  than 
the  two  left-hand  periods,  subtract,  and  to  the  difference  annex 
the  first  figure  of  the  next  period  for  a  second  dividend.  If  the 
cube  of  the  root  found  is  greater  than  the  two  left-hand  periods, 
diminish  the  second  term  of  the  root. 

5.  Take  three  times  the  square  of  the  two  terms  of  the  root 
found  for  a  second  trial  divisor,  and  the  quotient  for  the  third 
term  of  the  root.  Cube  the  three  terms  of  the  root  found,  and 
subtract  the  result  from  the  three  left-hand  periods,  and  continue 
the  operation  in  like  manner  until  all  the  terms  of  the  root  are 
found. 

Notes. — 1.  Tlie  quotient  obtained  by  dividing  the  dividend  by  the 
trial  divisor  may  be  too  large,  since  three  times  the  square  of  the  next 
figure  of  the  root  may  be  a  part  of  the  dividend.  Usually  the  term 
of  the  root  sought  is  the  quotient,  or  one  less  than  the  quotient. 

2.  When  a  dividend  does  not  contain  the  trial  divisor,  write  a 
cipher  for  the  next  term  of  the  root.  Take  three  times  the  square  of 
the  root  thus  formed  for  a  trial  divisor,  and  to  the  dividend  annex 
tlie  two  remaining  figures  of  the  period,  and  the  first  figure  of  the 
next  period  for  a  new  dividend. 

3.  If  the  number  is  not  a  perfect  cube,  the  root  may  be  approx- 
imated by  annexing  periods  of  decimal  ciphers,  thus  adding  decimal 
terms  to  the  root.  Sufficient  accuracy  is  usually  secured  by  continuing 
the  root  to  two  or  three  decimal  platfes. 

4.  When  both  terms  of  a  common  fraction  are  not  perfect  cubes, 
the  cube  root  may  be  found  approxiiilately  by  multiplying  both  terms 
of  the  fraction  by  the  square  of  the  denominator,  and  extracting  the 
root  of  the  resulting  fraction.  The  error  Avill  be  less  than  one  di- 
vided by  the  denominator  of  the  root. 

5.  The  above  methods  of  extracting  the  square  or  cube  root  of 
numbers,  is  a  general  method  by  Avhich  any  root  may  be  extracted. 
The  fourth  root,  for  example,  is  found  by  dividing  the  number  into 
periods  of  four  figures  each,  then  taking  the  fourth  root  of  the  left- 
hand  period  for  the  first  term  of  the  root,  four  times  the  cube  of  this 
first  term  for  a  trial  divisor,  and  the  remainder  with  the  first  term  of 
the  next  period  annexed,  for  a  dividend,  etc. 

6.  The  cube  root  of  a  perfect  cube  may  be  found  by  resolving  it 
into  its  prime  factors  and  taking  the  product  of  one  of  every  three  of 
those  that  are  equal. 


262 


COMPLETE  ARITHMETIC. 


Greoinetrical    Explanation    of   th.e    Process    of    Ex- 
tracting th.e  Cu-be  Root. 

417.  The  solid  contents  of  a  cube  are  found  by  cubing 
the  length  of  its  edge,  and,  con- 
versely, the  length  of  the  edge  is 
found  by  extracting  the  cube  root 
of  the  number  denoting  the  solid 
contents. 

Let  the  annexed  cut  represent 
a  cube  whose  solid  contents  are 
15625.  Required  the  length  of 
the  edge. 

Since  the  number  denoting  the  solid  contents  contains 
two  periods,  there  will  be  two 
terms  in  the  cube  root,  and 
since  the  greatest  cube  in  the 
left-hand  period  is  8,  the  tens' 
term  of  the  root  i.s  2  (Art.  414). 
Hence,  the  length  of  the  edge  of 
the  cube  is  20  plus  the  units' 
term  of  the  root. 
What  is  the  units'  term  ? 


Process. 

15625^25 

2^=_8_     6 

20^X3  =  1200)    7625 

20^X5X3=    6000 

20  X  52  X  3  =    1500 

53  =      125 

"  7625 


Taking  from  the  given  cube  a  cube  whose  edge  is  20  and 
whose  capacity  is  8000,  there  remains  a  solid  whose  capacity 


is  15625  —  8000,  which  is  7625.     An  inspection  of  the  an- 
nexed cut  shows  that  this  solid  contains  Hiree  equal  rectan- 


CUBE  ROOT. 


263 


gular  solids,  whose  inner  face  (20 2)  is  equal  to  the  face  of 
the  removed  cube  and  whose  thickness  equals  the  units' 
term  of  the  root.  What  is  the  thickness  of  each  of  these 
rectangular  solids? 

Since  they  compose  only  a  part  of  the  solid  whose  solid 
contents  are  7625,  their  thickness  can  not  be  greater  than 
the  quotient  obtained  by  dividing  7625  by  the  area  of  their 
joint  inner  faces,  which  is  20 ^  X  3,  or  1200.  The  quotient 
is  6,  which  is  at  least  one  greater  than  the  thickness  of  each 
of  the  three  rectangular  solids,  since  26^  is  greater  than 
15625,  the  solid  contents  of  the  given  cube.  Try  5  for  the 
thickness.  25^  =  15625,  and  hence  5  is  the  required  thick- 
ness, and  the  length  of  the  edge  of  the  given  cube  is  20  -j-  5, 
or  25. 

The  correctness  of  this  result  may  also  be  shown  by  find- 
ing the  solid  contents  of  the  several  parts  of  the  given  cube. 
The  solidity  of  the  cube  removed  is,  as  shown  above,  20^  =: 
8000.  The  joint  solidity  of  the  three  adjacent  rectangular 
solids  is  202  X  5  X  3  =  6000. 


Removing  these  three  rectan- 
gular solids,  there  remain  three 
other  rectangular  solids,  whose 
solidity  is  20  X  5^,  or  500  each, 
and  whose  combined  solidity  is 
500  X  3,  or  1500. 


Removing  these  three  rectangular  solids,  there  remains 
the  small  cube,  whose  solidity  is 
53  =  125.  fYpI 

Adding  the  solidity  of  the  sev- 
eral parts,  we  have  8000  +  6000 
-f  1500  +  125  =  15625,  which  is 
the  solidity  of  the  given  cube.  ,, :::::::". 

It  is  seen  that  the  cube  whose 
edge  is  20,  represents  the  cube  of 
the  tens  of  the  root;   the  three  ad- 


'V-' 


264^ 


COxMPLETE  ARITHMETIC. 


jacent  rectangular  solids  represent  three  times  the  product  of 
(lie  square  of  tens  by  the  units;  the  three  smaller  rectangular 
solids,  three  times  the  product  of  the  tens  by  the  square  of  the 
units;  and  the  smaller  cube,  tJie  cube  of  the  units. 


APPLICATIONS   OF  INVOLUTION  AND  EVOLU- 
TION. 


I.   TO   THE   KIGHT-ANGLED  TRIANGLE. 

418.  The  HypOtemise  of  a  right-angled  triangle  is 
the  side  opposite  the  right  angle.  The  other  two  sides  are 
called  the  Base  and  the  Perpendicidar.     (Art.  155.) 

419.  Principles. — 1.   The  square  of  the  hypotenuse  of  a 

right-angled  triangle  is  equal  to  the  sum 
of  the  squares  of  the  otJwr  two  sides. 

This  principle,  which  may  be  proven  by 
geometry,  is  illustrated  by  the  annexed 
diagram. 

2.  The  square  of  the  base  or  the  per- 
pendicular of  a  right-angled  triangle  is 
equal  to  the  square  of  the  hypotenuse  less 
the  square  of  the  other  side. 


(J 


PROBLEMS. 

1.  The  base  of  a  right-angled  triangle  is  8,  and  the  per- 
pendicular 6 :  what  is  the  length  of  the  hypotenuse  ? 

Solution. — Since  the  square  of  the  hypotenuse  equals  the  square 
of  the  base  plus  the  square  of  the  perpendicular,  the  hypotenuse 
equals  VS^ -]- 6^  =■- Vl^  =  10. 

2.  The  hypotenuse  of  a  right-angled  triangle  is  20  inches 
and  the  base  is  16  inches:  what  is  the  perpendicular? 

3.  The  hypotenuse  of  a  right-angled  triangle  is  45  feet 
and  the  perpendicular  is  27  feet :  what  is  the  base  ? 


APPLICATIONS  OF  INVOLUTION  AND  EVOLUTION.     265 

4.  A  rectangular  field  is  192  yards  long  and  144  yards 
wide :  what  is  the  lengtli  of  the  diagonal  ? 

5.  The  foot  of  a  ladder  is  18  feet  from  the  base  of  a 
building,  and  the  top  reaches  a  window  24  feet  from  the 
base :  what  is  the  length  of  the  ladder? 

6.  Two  boys  start  from  the  same  point,  and  one  walks  96 
rods  due  north,  and  the  other  72  rods  due  east :  how  far  are 
they  apart? 

7.  A  flag  pole  180  feet  high  casts  a  shadow  135  feet  in 
length :  what  is  the  distance  from  the  top  of  the  pole  to  the 
end  of  the  shadow? 

8.  A  boy  in  flying  his  kite  let  out  240  feet  of  string,  and 
the  distance  from  where  he  stood  to  a  point  directly  under 
the  kite  was  208  feet :  how  high  was  the  kite,  supposing  the 
string  to  be  straight? 

9.  A  rectangular  field  is  84  rods  long  and  63  rods  wide : 
what  is  the  side  of  a  square  field  of  the  same  area? 

10.  A  farm  is  125  rods  square,  and  a  rectangular  farm, 
containing  the  same  number  of  acres,  is  100  feet  in  length  : 
what  is  its  width  ? 

420.  Rules. — 1.  To  find  the  hypotenuse  of  a  right-angled 
triangle.  Extract  the  square  root  of  the  sum  of  the  squares  of 
the  other  two  sides. 

2.  To  find  the  base  or  the  perpendicular  of  a  right-angled 
triangle.  Extract  the  square  root  of  the  difference  between 
the  squares  of  the  hypotenuse  and  the  other  side. 

II.    TO    THE    CIRCLE. 

•     421.  Principles. — 1.    The  area  of  a  circle  is  equal  to  the 
square  of  its  diameter  midtiplied  by  .7854. 

2.  The  areas  of  two  circles  are  to  each  other  as  the  squares  of 
their  diameters. 

Note. — The  above  propositions  can  be  proven  -by  geometry.     The 
area  of  a  circle  also  equals  its  circumference  multiplied  by  one  fourth 
of  its  diameter.     (Art.  161.) 
C.Ar.— 23. 


266  COMPLETE  ARITHMETIC. 


PROBLEMS. 

11.  The  diameter  of  a  circle  is  15  inches :  what  is  its 
area? 

12.  A  circular  pond  is  100  feet  in  diameter :  how  many- 
square  yards  does  it  contain  ? 

13.  A  circular  room  has  an  area  of  78.54  square  yards: 
what  is  its  diameter? 

14.  How  many  circles,  each  3  inches  in  diameter,  will 
equal  in  area  a  circle  w^hose  diameter  is  2  feet? 

15.  How  many  circles,  each  15  inches  in  diameter,  will 
equal  in  area  a  circle  whose  diameter  is  5  feet? 

16.  A  horse,  tied  to  a  stake  by  a  rope,  can  graze  to  the 
distance  of  40  feet  from  the  stake:  on  how  much  surface 
can  it  graze  ? 

17.  A  horse,  tied  to  a  stake,  can  graze  on  218^  square 
yards  of  surface :  to  what  distance  from  the  stake  can  it 
graze  ? 

18.  How  many  circles,  each  3  inches  in  diameter,  contain 
the  same  area  as  a  surface  2.5  feet  square? 

III.    TO    THE    SPHERE. 

422,  Principles. — 1.  The  surface  of  a  sphere  is  equal  to 
the  square  of  the  diameter  multiplied  by  3.1416. 

2.  The  solidity  of  a  sphere  is  equal  to  the  cube  of  the  diameter 
multiplied  by  .5236. 

3.  Two  spheres  are  to  each  other  as  the  cubes  of  their  diame- 
ters. 

Note. — The  surface  of  a  sphere  may  also  he  found  bij  multiplying 
the  circumference  by  the  diameter;  and  the  solidity  by  multiplying  the 
surface  by  one  sixth  of  the  diameter.     (Arts.  474,  475.) 

PROBLEMS. 

19.  What  is  the  surface  of  a  sphere  whose  diameter  is  1 0 
inches  ? 

20.  How  many  square  miles  on  the  surface  of  the  earth, 
its  mean  diameter  being  7912  miles? 


APPLICATIONS  OF  INVOLUTION  AND  EVOLUTION.    267 

21.  How  many  cubic  miles  in  the  solidity  of  the  earth  ? 

22.  How  many  cubic  inches  in  a  cannon  ball  whose 
diameter  is  7  inches? 

23.  How  many  balls  2  inches  in  diameter,  equal  in  solid- 
ity a  ball  whose  diameter  is  8  inches? 

24.  The  diameter  of  the  earth  is  about  4  times  the  diam- 
eter of  the  moon :  how  many  times  larger  than  the  moon  is 
the  earth? 

25.  The  diameter  of  Jupiter,  the  largest  planet,  is  about 
85000  miles,  and  the  diameter  of  the  sun  is  about  850000 
miles :  how  many  times  larger  than  Jupiter  is  the  sun  ? 

26.  The  surface  of  the  planet  Mercury  contains  about 
28274400  square  miles :  what  is  its  diameter  ? 

27.  The  planet  Uranus  contains  about  18816613200000 
cubic  miles :  what  is  its  diameter  ? 

Suggestion.— Divide  the  solidity  by  .5236,  and  extract  the  cube 
root  of  the  quotient. 

28.  A  brass  ball  contains  904.7808  cubic  inches:  what  is 
its  diameter? 

29.  A  square  and  a  triangle  contain  an  equivalent  area, 
and  the  base  of  the  triangle  is  36.1  inches,  and  its  altitude 
is  5  inches :  what  is  the  side  of  the  square  ? 

30.  One  of  the  mammoth  pines  of  California  is  110  feet 
in  circumference :  what  is  its  diameter  ? 

31.  How  many  cubic  feet  in  a  portion  of  the  above  tree 
100  feet  in  length,  supposing  its  mean  circumference  to  be 
94ifeet? 

32.  The  mean  distance  of  the  earth  from  the  sun  (new 
value)  is  about  91400000  miles,  and  it  revolves  in  its  orbit 
in  365^  days :  what  is  its  mean  hourly  motion  ? 

33.  The  mean  distance  of  Mercury  from  the  sun  (new 
value)  is  about  35400000  miles,  and  it  revolves  in  its  orbit 
in  87.9  days:  what  is  its  mean  hourly  motion? 

34.  The  diameter  of  the  moon  is  about  2000  miles  :  how 
does  the  extent  of  the  moon's  surface  compare  with  that  of 
the  earth? 


268  COMPLETE  ARITHMETIC. 


SECTION  XVI. 


GENERAL   REVIEW. 

Note. — The  following  problems  are  selected  from  several  sets  used 
in  the  examination  of  pupils  for  promotion  to  high  schools,  and  in  the 
examination  of  teachers. 


MENTAL  PKOBLEMS. 

1.  If  3  apples  are  worth  2  oranges,  how  many  oranges 
are  24  apples  worth? 

2.  How  long  will  it  take  a  man  to  lay  up  $60,  if  he  earn 
$15  a  week  and  spend  $9? 

3.  i  of  74|  is  J  of  what  number? 

4.  "I  of  45  is  f  of  how  many  times  10? 

5.  A  has  20  cents ;  and  |  of  what  A  has  is  f  of  what 
B  has:  how  many  has  B? 

6.  If  -J  of  a  yard  of  cloth  cost  63  cents,  what  will  f  of 
a  yard  cost? 

7.  If  3  yards  of  muslin  cost  13|^  cents,  what  will  f  of  a 
yard  cost? 

8.  The  difference  between  f  and  |  of  a  number  is  10 : 
what  is  the  number? 

9.  What  fraction  is  as  much  greater  than  f  as  f  is  less? 

10.  A  piece  of  flannel  lost  |  of  its  length  by  shrinkage 
in  fulling,  and  then  measured  30  yards :  what  was  its  length 
before  fulling? 

11.  A  horse  cost  $90,  and  -j\  of  the  price  of  the  horse 
equals  f  of  3  times  the  cost  of  the  saddle :  what  did  the 
saddle  cost? 

12.  If  to  my  age  you  add  its  half,  its  third,  and  28  years, 
the  sum  will  be  three  times  ray  age :  what  is  my  age  ? 

13.  A  boy  being  asked  his  age,  said  that  f  of  80  was  f 
of  10  times  his  age:  what  was  his  age? 


GENERAL  REVIEW.  269 

14.  A  boy  gave  f  of  his  money  for  a  sled,  ^  of  it  for  a 
hat,  and  then  had  7  cents  left:  how  many  cents  had  he  at 
first? 

15.  I  of  my  money  is  in  my  purse,  |  in  my  hand,  and 
the  remainder,  which  is  25  cents,  is  in  my  pocket:  how 
much  money  have  I  ? 

16.  A  boy  having  f  of  a  dollar,  gave  |-  of  his  money  to 
John  and  ^  of  the  remainder  to  James :  what  part  of  a 
dollar  did  James  receive? 

17.  A  farmer  sold  f  of  his  sheep  and  then  bought  f  as 
many  as  he  had  left,  when  he  had  40  sheep :  hoAV  many  had 
he  at  first? 

18.  John  lost  f  of  his  money  and  spent  ^  of  the  re- 
mainder, and  then  had  only  10  cents:  how  much  money 
had  he  at  first? 

19.  A  man  sold  a  horse  for  $60,  which  was  |^  of  f  of  its 
cost:  how  much  was  lost  by  the  bargain? 

20.  A  man  sold  a  horse  for  $130,  which  was  f  more  than 
it  cost  him :  what  was  the  cost  of  the  horse  ? 

21.  A  sold  B  a  horse  for  -J-  more  than  its  cost,  and  B  sold 
it  for  $80,  losing  |-  of  its  cost :  how  much  did  A  pay  for  the 
horse  ? 

22.  At  $f  a  bushel,  how  many  bushels  of  corn  may  be 
bought  for  $8? 

23.  If  J  of  a  bushel  of  wheat  cost  $|,  what  part  of  a 
bushel  can  be  bought  for  $|? 

24.  If  $18f  will  purchase  f  of  a  load  of  corn,  what  part 
of  it  will  $16|  purchase? 

25.  If  2-}y  pounds  of  cheese  cost  3|  dimes,  what  part  of  a 
pound  can  be  bought  for  1  dime  ? 

26.  How  many  bushels  of  coal  at  12^^  cents  a  bushel  can 
be  bought  for  $15  ? 

27.  What  part  of  7  bushels  is  |  of  a  peck  ? 

28.  What  part  of  a  pound  of  gold  is  .25  of  an  ounce? 

29.  What  part  of  |  of  a  gallon  is  |  of  a  pint  ? 

30.  From  f  of  a  day  take  -J  of  an  hour. 

31.  If  a  staff  5  feet  long  cast  a  shadow  2  feet  long  at  12 


270  COMPLETE   ARITHMETIC. 

o'clock,  what  is  the  height  of  a  steeple  whose  shadow,  at  the 
same  hour,  is  80  feet? 

32.  If  a  five  cent  loaf  weigh  10  ounces  when  flour  is  $4 
a  barrel,  what  ought  it  to  weigh  when  flour  is  $5  a  barrel? 

33.  If  20  bushels  of  oats  will  feed  40  horses  80  days,  how 
long  will  180  bushels  feed  them  ? 

34.  If  a  horse  eat  2  bushels  of  oats  in  6  days,  in  how 
many  days  will  2  horses  eat  18  bushels  ? 

35.  If  3  men  can  mow  18  acres  of  grass  in  4  days,  how 
many  men  can  mow  9  acres  in  3  days? 

36.  A  garrison  of  20  men  is  supplied  with  provisions  for 
12  days :  if  12  men  leave,  how  long  will  the  provisions  serve 
the  remainder? 

37.  A  man  bought  a  watch  and  chain  for  $80,  and  the  chain 
cost  ^  as  much  as  the  watch:  how  much  did  each  cost? 

38.  A  has  1^  times  as  many  cents  as  B,  and  they  together 
have  40  cents  :  how  many  has  each  ? 

39.  A  pole  120  feet  high  fell  and  broke  into  two  parts, 
and  f  of  the  longer  part  was  equal  to  the  shorter :  how  long 
was  each  part? 

40.  A  and  B  together  own  824  sheep,  and  A  has  If  times 
as  many  as  B :  how  many  has  each  ? 

41.  A,  B,  and  C  rent  a  pasture  for  $42 ;  B  pays  half  as 
much  as  A,  and  C  half  as  much  as  B :  what  does  each 
pay? 

42.  A  and  B  own  a  farm  ;  A  owns  f  as  much  as  B,  and 
B  owns  40  acres  more  than  A:  how  many  acres  does  each 
own? 

43.  f  of  A's  money  is  |  of  B's,  and  f  of  B's  is  |  of  C's, 
which  is  $81 :  how  much  have  A  and  B  each  ? 

44.  If  a  man  can  reap  f  of  an  acre  of  wheat  in  a  day, 
how  much  can  6  men  reap  in  10  days? 

45.  A  makes  a  shoe  in  |  of  a  day ;  B  makes  one  in  f  of 
a  day:  how  many  shoes  can  both  make  in  a  day? 

46.  A  can  mow  an  acre  of  grass  in  f  of  a  day,  and  B  in 
•|  of  a  day :  how  long  will  it  take  both  together  to  mow  an 
acre  ? 


GENERAL  REVIEW.  271 

47.  A  can  mow  a  field  of  grass  in  5  days,  and  B  in  4  days: 
how  long  will  it  take  both,  working  together,  to  mow  it? 

48.  A  can  build  a  house  in  20  days,  but,  with  the  assist- 
ance of  C,  he  can  do  it  in  12  days :  in  what  time  can  C  do 
it  alone  ? 

49.  A  alone  can  build  a  certain  wall  in  6  days,  B  alone 
in  10  days,  and  C  alone  in  15  days :  in  how  many  days  can 
they  all  together  build  it? 

50.  A,  B,  and  C  can  do  a  job  in  20  days ;  A  and  B  can 
do  it  in  40  days ;  and  A  and  C  in  30'  days :  in  how  many 
days  can  each  do  it  alone  ? 

51.  A  broker  bought  rail-road  stock  at  80  and  sold  it  at 
70 :  what  per  cent,  did  he  lose  ? 

52.  A  broker  bought  stock  at  70  and  sold  it  at  90 :  what 
per  cent,  did  he  gain  ? 

53.  A  merchant  bought  40  yards  of  cloth  for  $90  :  at  how 
much  a  yard  must  he  sell  it  to  gain  33^  per  cent.  ? 

54.  For  how  much  must  tea  costing  90  cents,  be  sold  to 
gain  12^  per  cent.  ? 

55.  A  man  bought  a  hat  for  $5  and  sold  it  for  $6  :  what 
per  cent,  did  he  gain  ? 

56.  I  sell  cloth  at  $2.50  a  yard  and  gain  25  per  cent.; 
for  how  much  must  I  sell  it  to  lose  20  per  cent.  ? 

57.  A  man  earned  a  certain  sum  of  money,  and,  after 
adding  to  it  $12.50,  found  that  what  he  then  had  was  133J 
per  cent,  of  what  he  earned :  how  much  did  he  earn  ? 

58.  A  man  sold  a  watch  for  $90,  and  gained  50  per  cent. : 
what  per  cent,  would  he  have  gained  if  he  had  sold  it  for 
$75? 

59.  f  of  the  price  received  for  an  article  is  equal  to  f  of 
its  cost :  what  is  the  gain  per  cent.  ? 

60.  Two  men,  A  and  B,  engaged  in  trade  with  different 
capitals ;  A  lost  33^  per  cent,  of  his  capital,  and  B  gained 
50  per  cent,  on  his,  when  each  had  $600:  with  what  capital 
did  each  begin  trade? 

61.  How  much  grain  must  I  take  to  mill  to  bring  away 
2  bushels  after  the  miller  has  taken  10  per  cent,  for  toll  ? 

J 


272  COMPLETE  ARITHMETIC. 

62.  At  what  rate  per  cent. ,  simple  interest,  will  $1  double 
itself  in  8  years? 

63.  The  interest  on  a  certain  sum  for  4  years  was  ^  the 
sum :  what  was  the  rate  per  cent.  ? 

64.  Two  men  start  from  two  places  495  miles  apart,  and 
travel  toward  each  other ;  one  travels  20  miles-  a  day,  and 
the  other  25  miles  a  day :  in  how  many  days  will  they 
meet? 

65.  A  owes  f  of  B's  income,  but,  by  saving  -^  of  B's 
income  annually,  he  can  pay  his  debt  in  5  years,  and  have 
$50  left:  what  is  B's  income? 

66.  C  and  D  are  traveling  in  the  same  direction  ;  C  is 
18  miles  ahead  of  D,  but  D  travels  7  miles  while  C  travels 

4  :  how  many  miles  from  the  place  of  starting  will  D  have 
traveled  when  he  overtakes  C? 

67.  If  a  man  traveling  14  hours  a  day,  perform  half  a 
journey  in  5  days,  how  long  will  it  take  to  perform  the 
other  half,  if  he  travel  10  hours  a  day  ? 

68.  If  a  man  can  do  a  piece  of  w^ork  in  9f  days,  working 
8  hours  a  day,  how  long  will  it  take,  if  he  work  6  hours  a 
day? 

69.  A  is  20  years  old ;  the  sum  of  the  ages  of  B  and  C 
equals  4  times  A's  age ;  C's  age  is  ^  of  A's  and  B's  to- 
gether :  what  is  the  age  of  each  ? 

70.  A  hare  is  30  rods  before  a  hound,  but  the  hound  runs 
7  rods  while  the  hare  runs  5 :  how  far  must  the  hound  run 
to  catch  the  hare? 

71.  A  hare  starts  50  leaps  before  a  hound,  and  leaps  4 
times  while  the  hound  leaps  3  times ;  but  2  of  the  hound's 
leaps  equal  4  of  the  hare's  :  how  many  leaps  must  the  hound 
take  to  catch  the  hare? 

72.  If  a  steamer  sails  9  miles  an  hour  down  stream,  and 

5  miles  an  hour  up  stream,   how  far  can  it  go  down  stream 
and  back  again  in  14  hours? 

73.  A  steamer  sails  a  mile  down  stream  in  5  minutes, 
and  a  mile  up  stream  in  7  minutes :  how  far  down  stream 
can  it  go  and  return  again  in  one  hour? 


GENERAL  REVIEW.  273 

74.  A  pipe  will  fill  a  cistern  in  4  hours,  and  another 
will  empty  it  in  6  hours :  how  long  will  it  take  to  fill  it 
when  both  pipes  run  ? 

75.  At  what  time  between  six  and  seven  o'clock  are  the 
hour  and  minute  hands  of  a  clock  together  ?  ^ 

WRITTEN  PROBLEMS. 

76.  The  minuend  is  1250,  and  the  remainder  592 :  what 
is  the  subtrahend  ? 

77.  The  quotient  is  71,  the  divisor  42,  and  the  remainder 
15  :  what  is  the  dividend  ? 

78.  If  a  certain  number  be  multiplied  by  22,  and  64  be 
subtracted  from  the  product,  and  the  remainder  be  divided 
by  4,  the  quotient  will  be  50:  what  is  the  number? 

79.  What  will  be  the  cost  of  3760  lbs.  of  hay,  at  $8.50  a 
ton? 

80.  At  $24.50  per  acre,  how  many  acres  of  land  can  be 
bought  for  $3560.75? 

81.  Add  I,  I,  1  of  I,  and  |  of  2i 

82.  From  17^  take  f  of  6 J,  and  multiply  the  remainder 

by  |. 

83.  Multiply  f  of  f  by  |^  of  f,  and  divide  the  product 

by  A. 

84.  Divide  f  of  6-1  by  |  of  7^. 

85.  What  number  multiplied  by  28f  will  produce  145? 

86.  From  the  sum  of  215|  and  125 J  take  their  differ- 
ence. 

87.  Multiply  f  -f  I  of  -I  by  I  - 1  of  f . 

88.  Divide  2^  by  3^,  and  multiply  the  quotient  by  3J. 

89.  What  must  8|f  be  multiplied  by  that  the  product 
may  be  3? 

90.  A  man  bought  ^^  of  a  section  of  land  for  $2880,  and 
sold  f  of  it  at  $10  an  acre,  and  the  rest  at  $12  an  acre: 
how  much  did  he  gain? 

91.  A  merchant  owning  |-  of  a  ship  sells  -f  of  his  share 
for  $16800 :  what  is  the  value  of  the  whole  ship,  at  this 
rate,  and  what  part  of  the  ship  has  he  left? 


274  COMPLETE   ARITHMETIC. 

92.  Add  9  thousandths,  3  hundredths,  and  7  units. 

93.  From  15  ten-thousandths  take  27  milliouths,  and 
multiply  the  difference  by  20.5. 

94.  Multiply  160  by  .016,  and  divide  the  product  by 
.0025.     - 

95.  Multiply  15  thousandths  by  15  hundredths,  and  from 
the  product  take  15  millionths. 

96.  Divide  256  thousandths  by  16  millionths. 

97.  Multiply  625  by  .003,  and  divide  the  result  by  25. 

98.  Change  ^|^  to  a  decimal,  and  divide  the  result  by  2^. 

99.  Change  yf^-  to  a  decimal,  and  divide  the  result  by 
5000. 

100.  Reduce  .625  of  a  pound  Troy  to  lower  integers. 

101.  What  decimal  of  a  rod  is  .165  of  a  foot? 

102.  What  will  63  thousandths  of  a  cord  of  wood  cost, 
at  S2.25  per  cord? 

103.  How  many  minutes  will  there  be  in  the  month  of 
February,  1880? 

104.  How  many  seconds  are  there  in  the  three  summer 
months  ? 

105.  How  many  steps,  2  ft.  4  in.  each,  will  a  person 
take  in  walking  2\  miles? 

106.  How  many  times  will  a  wheel,  12  ft.  6  in.  in  cir- 
cumference, turn  round  in  rolling  one  mile? 

107.  How  many  acres  in  a  street  4  rods  wide  and  2J 
miles  long? 

108.  How  many  yards  of  carpeting,  f  of  a  yard  wide, 
will  it  take  to  cover  a  parlor,  18^  feet  long  and  15  feet 
wide  ? 

109.  How  many  grains  in  14  ingots  of  silver,  each  weigh- 
ing 27  oz.  10  pwt.? 

110.  How  many  square  feet  of  lumber  in  40  boards,  each 
12  feet  long  and  7^  inches  wide? 

111.  AVhat  will  a  board  20  feet  long  and  9  inches  wide 
cost,  at  $30  a  thousand? 

112.  What  will  it  cost  to  lay  a  pavement  36  feet  long 
and  9  feet  6  inches  wide,  at  40  cents  a  square  yard? 


GENERAL  REVIEW.  275 

113.  A  pile  of  wood,  containing  10  cords,  is  20  feet  long 
and  8  feet  wide :  how  high  is  it  ? 

114.  What  is  the  value  of  a  pile  of  wood  40  feet  long,  8 
feet  wide,  and  5^  feet  high,  at  $5.30  a  cord? 

115.  How  many  sacks,  holding  2  bu.  3  pk.  2  qt.  each, 
can  be  filled  from  a  bin  containing  366  bu.  3  pk.  4  qt.  of 
wheat  ? 

116.  A  lady  bought  6  silver  spoons,  each  weighing  3  oz. 
3  pwt.  8  gr.,  at  $2.25  an  ounce,  and  a  gold  chain,  weighing 
14  pwt.,  at  $1.25  a  pwt. :  what  was  the  cost  of  both  spoons 
and  chain? 

117.  How  many  bricks  will  it  require  to  build  a  wall  2 
rods  long,  6  feet  high,  and  18  inches  thick,  each  brick  being 
8  inches  long,  4  inches  Avide,  and  2^  inches  thick? 

118.  Cincinnati  is  7°  49'  west  of  Baltimore:  when  it  is 
noon  at  Baltimore,  what  is  the  hour  at  Cincinnati? 

119.  New  York  is  75  degrees  of  longitude  west  of  Lon- 
don :  when  it  is  noon  at  New  York  what  is  the  hour  at 
London  ? 

120.  Boston  is  71°  4'  9"  W.  longitude,  and  Cleveland  is 
81°  47'  W. :  when  it  is  4  P.  M.  at  Cleveland,  what  is  the 
hour  at  Boston  ? 

121.  What  part  of  a  rod  is  2  ft.  9  in.? 

122.  Reduce  5  fur.  8  rd.  to  the  decimal  of  a  mile. 

123.  Reduce  f  of  a  square  yard  to  the  fraction  of  an 
acre. 

124.  From  |  of  a  pound  Troy  take  f  of  an  ounce. 

125.  Reduce  f  of  a  quart  to  the  fraction  of  a  bushel. 

126.  A  regiment  lost  8  %  of  its  men  in  a  battle,  and 
25%  of  those  that  remained  died  from  sickness,  and  it  then 
mustered  621  men:  how  many  men  were  in  the  regiment  at 
first? 

127.  A  quantity  of  sugar  was  bought  for  $150,  and  sold 
for  $167.50:  what  was  the  gain  per  cent.? 

128.  A  merchant  bought  500  yards  of  cloth  for  $1800 : 
for  how  much  a  yard  must  he  sell  it  to  gain  25  %  ? 

129.  A  man  sold  a  piece  of  cloth  for  $24,  and  thereby 


276  COMPLETE  ARITHMETIC. 

lost  25  %  ;  if  he  had  sold  it  for  $34,  wouhl  he  have  gained 
or  lost,  and  what  per  cent.  ? 

130.  I  sold  goods  at  20  ^  gain,  and,  investing  the  pro- 
ceeds, sold  at  20  %  loss :  did  I  gain  or  lose  by  the  opera- 
tion, and  what  per  cent.  ? 

131.  Sold  2  carriages,  at  $240  apiece,  and  gained  20  % 
on  one  and  lost  20  %  on  the  other :  how  much  did  I  gain 
or  lose  in  the  transaction? 

132.  A  man  bought  a  horse  for  $72,  and  sold  it  for  25  % 
more  than  cost,  and  10  %  less  than  he  asked  for  it :  what 
did  he  ask  for  it? 

133.  A  merchant  marked  a  lot  of  goods,  costing  $5800, 
at  30  %  above  cost,  but  sold  them  at  10  %  less  than  the 
marked  price :  how  much  and  what  per  cent,  did  he 
gain  ? 

134.  What  must  I  ask  for  cloth,  costing  $4  a  yard,  that 
I  may  deduct  20  %  from  my  asking  price  and  still  make 
20%? 

135.  A  man  bought  stock  at  25  %  below  par  and  sold  it 
at  20  %  above  par :  what  per  cent,  did  he  make  ? 

136.  A  fruit  dealer  lost  33J  per  cent,  of  a  lot  of  apples, 
and  sold  the  remainder  at  a  gain  of  50  ]3er  cent.  :  required 
the  per  cent,  of  gain  or  loss. 

137.  I  bought  63  kegs  of  nails,  each  keg  containing 
100  lbs.,  at  4|-  cents  a  pound,  and  sold  \  of  them  for  what 
\  of  them  cost:  what  per  cent,  did  I  lose  on  the  part  sold? 

138.  I  bought  $128.25  worth  of  goods;  kept  them  on 
hand  6  months  when  money  was  worth  8  %  interest,  and 
then  sold  them  at  a  net  gain  of  6  % :  for  how  much  were 
they  sold? 

139.  When  money  was  worth  9%  interest,  I  bought  $800 
worth  of  goods,  kept  them  4  months  and  then  sold  them 
for  $959. 10 :  what  per  cent,  on  the  cost  did  I  gain  ? 

140.  A  house  valued  at  $3240  is  insured  for  |  of  its 
value,  at  f  %  :  what  is  the  premium  ? 

141.  I  pay  $19.20  premium  for  insuring  my  house  for  \ 
of  its  value,  at  1^  %  '-  what  is  the  value  of  my  house? 


GENERAL  REVIEW.  277 

142.  A  capitalist  sent  a  broker  $25000  to  invest  in  cotton, 
after  deducting  his  commission  of  2^%:  how  much  cotton, 
at  5  cents  a  pound,  did  the  broker  purchase  ? 

143.  x\n  agent  received  $502.50  to  purchase  cloth,  after 
deducting  |  %  commission  :  how  many  yards  did  he  buy 
at  $1.25  a  yard? 

144.  What  is  the  interest  of  $125.50  for  7  months  and 
10  days,  at  7  %  ? 

145.  What  is  the  interest  of  $50000  for  one  day,  at  8%? 

146.  What  is  the  interest  of  $75.50  from  June  12,  1869, 
to  Aug.  6,  1870,  at  71  %  ? 

147.  A  man  loaned  $800  for  2  years  and  6  months,  and 
received  $90  interest :  what  was  the  rate  per  cent.  ? 

148.  At  what  rate  per  cent,  will  $311.50  amount  to 
$337.40  in  1  yr.  4  mo.  ? 

149.  What  sum  of  money  will  yield  as  much  interest  in 
3  years,  at  4^  per  cent.,  as  $540  yields  in  1  yr.  8  mo.,  at 
7%? 

150.  The  amount  of  a  certain  principal-  for  3  years,  at  a 
certain  rate  per  cent.,  is  $750,  and  for  5  years  $1250:  what 
is  the  principal,  and  what  is  the  rate  per  cent.  ? 

151.  A  note  for  $500,  dated  Oct.  8,  1864,  and  bearing 
interest  at  9  %,  is  indorsed  as  follows:  Nov.  4,  1865,  $30; 
Jan.  30,  1866,  $250.     What  was  due  July  1,  1866? 

152.  What  is  the  present  worth  of  a  note  of  $1320,  due 
in  3  years  and  4  months,  without  interest,  money  being 
worth  6  %  ?     What  is  the  discount  ? 

153.  What  is  the  true  discount  of  $236,  due  in  3  years, 
at  6  %  ? 

154.  What  is  the  bank  discount  on  $125,  j)ayable  in  90 
days,  at  8  %  ? 

155.  AVhat  is  the  difference  between  the  true  discount 
and  the  bank  discount  of  $359.50,  for  90  days,  without 
grace,  at  12%? 

156.  For  what  sum  must  I  give  my  note  at  a  bank,  pay- 
able in  4  months,  at  10^,  to  get  $300? 

157.  I  borrow  of  A  $150  for  6  months,  and  afterward  I 


278  COMPLETE  ARITHMETIC. 

lend  him  $100:  how  long  may  he  keep  it  to  balance  the 
use  of  the  sum  he  lent  me? 

158.  A  owes  B  $800,  of  which  $50  is  due  in  2  months, 
$100  in  5  months,  and  the  remainder  in  8  months :  what  is 
the  equated  time  for  the  whole  sum? 

159.  A  man  owes  $300  due  in  5  months,  and  $700  due  in 
3  months,  and  $200  due  in  8  months :  if  he  pays  ^  of  the 
whole  in  2  months,  when  ought  the  other  half  to  be  paid? 

160.  I  have  sold  50  bushels  of  wheat  for  A,  and  60 
bushels  for  B,  receiving  $150  for  both  lots:  if  A's  wheat  is 
worth  20  %  more  than  B's,  how  much  ought  I  to  pay  each  ? 

161.  Two  men  divided  a  lot  of  wood,  which  they  pur- 
chased together  for  $27;  one  took  5^  cords,  the  other  8: 
what  ought  each  to  pay? 

162.  If  8  men  cut  84  cords  of  wood  in  12  days,  working 
7  hours  a  day,  how  many  men  will  cut  150  cords  in  10 
days,  working  5  hours  a  day? 

163.  If  16  horses  consume  84  bushels  of  grain  in  24  days, 
how  many  bushels  of  grain  will  supply  36  horses  16  days? 

164.  If  the  wages  of  24  men  for  4  days  are  $192,  what 
will  be  the  wages  of  36  men  for  3  days? 

165.  If  4  men  in  7f  days  earn  $53f ,  how  much  will  7 
men  earn  in  :^  of  a  day  ? 

166.  A  and  B  traded  in  company  and  gained  $750,  of 
which  B's  share  was  $600;  A's  stock  was  $1200:  what  was 
B's  stock? 

167.  A  and  B  formed  a  partnership  for  1  year,  and  A 
put  in  $2000  and  B  $800 :  how  much  more  must  B  put  in 
at  the  close  of  6  months  to  receive  one-half  of  the  profits? 

168.  A  and  B  engage  in  trade;  A  puts  in  $200  for  5 
months,  B  $300  for  2  months ;  they  draw  out  capital  and 
profits  to  the  amount  of  $1389:  what  was  each  man's  share? 

169.  What  is  the  square  root  of  41616?     Of  420.25? 

170.  What  is  the  cube  root  of  46656?     Of  42.875? 

171.  A  certain  window  is  30  feet  from  the  ground:  how 
far  from  the  base  of  the  building  must  the  foot  of  a  ladder 
50  feet  long   be  placed  to  reach  the  window? 


GENERAL  REVIEW.  279 

172.  Two  men  start  from  the  same  point;  one  travels 
52  miles  north  and  the  other  39  miles  west:  how  far  are 
they  apart? 

173.  A  house  is  40  feet  high  from  the  ground  to  the  eaves, 
and  it  is  required  to  find  the  length  of  a  ladder  which  will 
reach  the  eaves,  supposing  the  foot  of  the  ladder  can  not  be 
placed  nearer  to  the  house  than  30  feet. 

174.  How  many  rods  of  fence  will  inclose  10  acres  in  the 
form  of  a  square  ? 

175.  A  floor  is  24  feet  long  and  15  feet  wide:  what  is 
the  distance  between  tw^o  opposite  corners? 

176.  A  room  is  20  feet  long,  16  feet  wide,  and  12  feet 
high :  what  is  the  distance  from  one  of  the  lower  corners  to 
the  upper  opposite  corner? 

177.  How  many  cubic  feet  in  a  stone  8  feet  long,  5^  feet 
wide,  and  3^  feet  thick? 

178.  How  many  square  feet  on  the  surface  of  a  stone  6 
feet  long,  4  feet  wide,  and  H  feet  thick? 

179.  There  is  a  circular  field  40  rods  in  diameter :  w^hat 
is  its  circumference?     How  many  acres  does  it  contain? 

180.  The  area  of  a  circle  is  470.8f  square  inches :  what 
is  the  length  of  its  diameter? 

181.  How  many  iron  balls  2  inches  in  diameter,  will 
weigh  as  much  as  an  iron  ball  8  inches  in  diameter? 

182.  How  many  cubical  blocks,  each  edge  of  which  is  -J- 
of  a  foot,  are  equivalent  to  a  block  of  wood  8  feet  long,  4 
feet  wide,  and  2  feet  thick? 

183.  How  many  bushels  of  wheat  will  fill  a  bin  8  feet 
long,  5  feet  wide,  and  4  feet  deep? 

184.  How  many  gallons  of  w^ater  will  a  cistern  contain 
which  is  7  feet  long,  6  feet  wide,  and  11  feet  deep? 

185.  How  many  gallons  of  water  will  fill  a  circular  cis- 
tern 6  feet  deep  and  4  feet  in  diameter? 

186.  Divide  $1000  among  A,  B,  and  C,  and  give  A  $120 
more  than  C,  and  C  $95  more  than  B. 

187.  A  can  mow  2  acres  in  3  days,  and  B  5  acres  in  6 
days :  in  how  many  days  can  they  together  mow  9  acres  ? 


280  COMPLETE  ARITHMETIC. 

188.  A  sold  cloth  to  B  and  gained  10  per  cent.  ;  B  sold 
it  to  C  and  gained  10  per  cent.  ;  C  sold  it  to  D  for  $726 
and  gained  1 0  per  cent.  :  how  much  did  it  cost  A  ? 

189.  A  man  steps  2  feet  8  inches,  and  a  boy  1  foot  10 
inches;  but  the  boy  takes  8  steps  while  the  man  takes  5: 
how  far  will  the  boy  walk  while  the  man  walks  3 J  miles  ? 

190.  A  father  bequeathed  j\  of  his  estate  to  his  eldest 
son,  f  of  the  remainder  to  his  second  son,  and  the  rest  to 
his  youngest  son ;  by  this  arrangement  the  eldest's  share 
was  $1300  more  than  the  youngest's:  what  was  the  share 
of  each  son  ? 

191.  If  7  bushels  of  wheat  are  worth  10  bushels  of  rye, 
and  5  bushels  of  rye  are  worth  14  bushels  of  oats,  and 
6  bushels  of  oats  are  Avorth  $3.13,  how  many  bushels  of 
wheat  will  $50  buy? 

192.  In  a  company  of  90  persons  there  are  4  more  men 
than  women  and  10  more  children  than  men  and  women 
together :  how  many  of  each  in  the  company  ? 

193.  Divide  $630  among  3  persons  so  that  the  second 
shall  have  f  as  much  as  the  first,  and  the  third  -^  as  much 
as  the  other  two  together. 

194.  A  and  B  can  do  a  piece  of  work  in  12  days,  B  and 
C  in  9  days,  and  A  and  C  in  6  days :  how  long  will  it  take 
each  alone  to  do  it? 

195.  A  and  B  perform  together  -^j^  of  a  piece  of  work  in 
2  days,  when,  B  leaving,  A  completes  it  in  ^  a  day:  in 
what  time  can  each  do  it  alone? 

196.  C  and  D  engage  in  trade  with  different  sums  of 
money ;  C  loses  40  per  cent,  of  his  capital,  and  D  gains  57 
per  cent,  on  his,  when  their  capitals  are  equal :  how  much 
greater  was  C's  capital  than  D's  when  they  began  business  ? 

197.  A  man  walks  100  miles  in  2  days,  and  ^  of  the 
distance  walked  the  first  day,  added  to  ^  the  distance  walked 
the  second  day,  equals  half  the  distance  walked  the  first 
day :   how  far  did  he  walk  each  day  ? 

198.  How  far  from  the  end  of  a  stick  of  timber  30  feet 
long,  of  equal  size  from  end  to  end,  must  a  handspike  be 


TEST  QUESTIONS.  281 

placed  so  that  3  men,  2  at  the  handspike  and  1  at  the  end 
of  the  stick,  may  each  carry  ^  of  its  weight  ? 

199.  Two  trees  stand  on  opposite  sides  of  a  stream  40  feet 
wide ;  the  height  of  one  tree  is  to  the  width  of  the  stream 
as  8  is  to  4,  and  the  width  of  the  stream  is  to  the  height  of  the 
other  as  4  is  to  5:  what  is  the  distance  between  their  tops? 

200.  A  cistern  is  filled  by  two  pipes,  one  of  which  will 
fill  it  in  two  hours,  and  the  other  in  3  hours ;  it  is  emptied 
by  three  pipes,  the  first  of  which  will  empty  it  in  5  hours, 
the  second  in  6  hours,  and  the  third  in  7^  hours :  if  all  the 
pipes  be  left  open,  in  what  time  will  it  be  filled? 

TEST   QUESTIONS. 

1.  What  is  a  number?  In  how  many  ways  may  numbers  be  repre- 
sented?    Name  them. 

2.  What  is  the  difference  between  numeration  and  notation?  Be- 
tween the  Arabic  notation  and  the  Roman?  Between  the  English 
numeration  and  the  French  ? 

3.  What  is  the  simple  value  of  the  figure  5  in  452?  What  is  its 
local  value?  How  is  the  local  value  of  a  figure  affected  by  its  re- 
moval one  order  to  the  left?  One  order  to  the  right?  How  is  the 
value  of  a  number  affected  by  annexing  a  cipher?     Why? 

4.  How  many  units  are  there  in  the  sum  of  two  or  more  integers? 
Why  in  addition  are  like  orders  of  figures  written  in  the  same  column  ? 
Why  in  adding  numbers  do  we  begin  at  the  right  hand? 

5.  Why  are  the  minuend,  subtrahend,  and  difference  like  numbers? 
Show  that  the  adding  of  10  to  a  term  of  the  minuend,  and  1  to  the 
next  higher  term  of  the  subtrahend,  increases  the  minuend  and  sub- 
trahend equally. 

6.  Why  must  the  multiplier  be  an  abstract  number?  When  the 
multiplicand  is  concrete,  what  is  true  of  the  product  ?  Why  ?  How 
is  the  product  affected  by  dividing  either  the  multiplicand  or  the 
multiplier?  Show  that  annexing  two  ciphers  to  an  integer,  and 
dividing  the  result  by  4,  gives  the  product  of  the  number  multiplied 
by  25. 

7.  What  kind  of  number  is  the  quotient  when  both  divisor  and 
dividend  are  like  numbers?  When  the  divisor  is  abstract  and  the 
dividend  concrete?  What  is  the  difference  between  short  division 
and  long  division?  Of  what  order  is  the  first  left-hand  figure  of  the 
quotient? 

C.Ar.-24. 


282  COMPLETE  ARITHMETIC. 

8.  How  is  the  quotient  affected  by  multiplying  or  dividing  both 
dividend  and  divisor  by  the  same  number?  By  multiplying  the  divi- 
dend by  any  number  greater  than  1  ?  On  what  principle  may  the  four 
fundamental  rules  be  reduced  to  two? 

9.  Name  all  the  prime  numbers  from  1  to  20  inclusive.  Show  that 
two  composite  numbers  may  be  prime  with  respect  to  each  other. 
What  is  meant  by  the  factors  of  a  number?  The  prime  factors? 
Show  that  the  common  factor  of  two  or  more  numbers  is  a  factor  of 
their  sum. 

10.  Why  is  the  factor  of  a  number  its  divisor?  How  is  a  number 
affected  by  the  canceling  of  a  factor?  On  what  principle  may  the 
common  factors  of  a  dividend  and  a  divisor  be  canceled  ? 

11.  When  is  a  divisor  a  common  divisor?  What  is  the  greatest 
common  divisor  of  two  or  more  numbers?  Show  that  the  common 
divisor  of  two  numbers  is  a  divisor  of  their  sum  and  diff'erence.  In 
how  many  ways  may  the  greatest  common  divisor  of  two  or  more 
numbers  be  found? 

12.  How  many  multiples  has  every  number?  What  is  a  common 
multiple?  What  is  the  least  common  multiple  of  two  or  more 
numbers?  In  how  many  ways  may  the  least  common  multiple  of 
two  or  more  numbers  be  found  ? 

13.  What  is  the  difference  between  a  divisor  and  a  multiple  of  a 
number  ?  Between  the  terms  factor,  divisor,  and  measure  ?  Is  2i  a 
divisor  of  5?     Is  12J  a  multiple  of  5?     Is  12}  a  multiple  of  6]-? 

14.  What  is  a  fraction?  In  what  two  ways  may  a  fi'action  be  ex- 
pressed? When  a  fraction  is  expressed  by  words,  which  word  or 
words  denote  the  denominator? 

15.  What  is  the  difference  between  the  unit  of  a  fraction  and  a  frac- 
tional unit?  Which  term  of  a  fraction  denotes  the  size  of  the  fractional 
unit?  When  is  the  value  of  a  fraction  equal  to  1?  Greater  than  1? 
Less  than  1  ? 

16.  Show  that  the  division  or  multiplication  of  both  terms  of  a 
fraction  by  the  same  number,  does  not  change  its  value.  How  is  the 
value  of  a  proper  fraction  affected  by  adding  the  same  number  to  both 
of  its  terms?   By  subtracting  the  same  number  from  both  of  its  terms? 

17.  On  what  principle  is  a  fraction  reduced  to  lower  terms?  To 
higher  terms?  On  what  principle  are  two  or  more  fractions  reduced 
to  a  common  denominator? 

18.  In  what  two  ways  may  a  fraction  be  multiplied  by  an  integer? 
Why?  In  what  two  ways  may  a  fraction  be  divided  by  an  integer? 
In  what  three  ways  may  a  fraction  be  divided  by  a  fraction?  Wiiy 
must  fractions  have  a  common  denominator  before  they  can  be  added 
or  subtracted  ? 


TEST  QUESTIONS.  283 

19.  What  is  a  decimal  fraction?  Is  the  fraction  fifteen-hundredtks  a 
decimal  fraction?  In  Avhat  two  Avays  may  it  be  expressed  by  figures? 
Which  is  called  the  decimal  form?  What  is  the  denominator  of  a 
decimal  fraction? 

20.  What  is  meant  by  decimal  places?  What  is  the  name  of  the 
third  decimal  order  from  units?  The  sixth?  The  ninth?  How  is 
a  decimal  read  ? 

21.  How  is  the  local  value  of  a  decimal  figure  affected  by  its  re- 
moval one  order  to  the  right  ?  One  order  to  the  left  ?  How  is  the 
value  of  a  decimal  affected  by  annexing  decimal  ciphers?  Why? 
By  prefixing  decimal  ciphers?     Why? 

22.  Plow  is  a  decimal  reduced  to  a  common  fraction?  A  common 
fraction  to  a  decimal?  Wliy  can  decimals  be  added  and  subtracted 
like  integers? 

23.  Why  does  the  product  contain  as  many  decimal  places  as  both 
multiplicand  and  multiplier?  Why  does  the  dividend  contain  as 
many  decimal  places  as  both  divisor  and  quotient? 

24.  How  is  a  decimal  divided  by  10,  100,  etc.?  How  is  a  decimal 
multiplied  by  10,  100,  etc.?  Why  are  numbers  denoting  sums  of 
money  added  and  subtracted  like  decimals? 

25.  What  is  a  rectangle?  How  is  its  area  found ?  What  is  a  circle? 
How  is  its  area  found? 

26.  What  is  a  right-angled  triangle ?     How  is  its  area  found? 

27.  What  is  a  rectangular  solid  ?  What  is  the  difference  between 
an  edge  and  a  face  of  a  solid? 

28.  Show  that  the  product  of  the  three  dimensions  of  a  rectangular 
solid  represents  its  volume  or  solid  contents.  How  are  the  contents  of 
a  cylinder  found? 

29.  Is  every  concrete  number  denominate?  Give  examples.  What 
is  the  difference  between  a  simple  denominate  number  and  a  compound 
number?     Give  examples. 

30.  What  do  denominate  numbers  express  ?  What  is  the  difference 
between  reduction  descending  and  reduction  ascending? 

31.  How  are  denominate  fractions  reduced  from  a  higher  to  a  lower 
denomination?  From  a  lower  to  a  higher?  How  is  a  denominate 
number  reduced  to  the  fraction  of  a  higher  denomination  ?  Give  an 
example. 

32.  What  is  the  Metric  System  ?  What  is  the  primary  unit  of  the 
system?  What  is  its  length  in  inches?  What  is  a  liter?  What  is  a 
gram. 

33.  How  are  the  multiples  of  the  meter,  liter,  and  gram  named  ? 
How  are  the  subdivisions  named  ? 

34.  What  is  the  difference  between  simple  addition  and  compound 
addition?     In  what  respect  are  the  processes  alike? 


284  COMPLETE  ARITHMETIC. 

35.  When  are  compound  numbers  of  the  same  kind  ?  Give  exam- 
ples. How  is  a  compound  number  divided  by  another  of  the  same 
kind? 

36.  What  part  of  tlie  equator  passes  beneath  the  vertical  rays  of  the 
Bun  every  hour?  What  part  of  the  tropic  of  Cancer?  What  part  of 
any  parallel  situated  between  the  polar  circles? 

37.  Why  is  the  time  of  day  earlier  at  New  York  than  at  St.  Louis  ? 
When  the  difference  in  longitude  between  two  places  is  given,  how  is 
the  difference  in  time  found? 

38.  What  is  meant  by  5  per  cent,  of  a  number?  What  is  the  dif- 
ference between  the  terms  rate  per  cent,  and  rate?     Give  examples. 

39.  What  four  numbers  are  considered  in  percentage?  Define  each. 
Give  the  four  cases  of  percentage  and  the  formula  for  each. 

40.  What  is  the  difference  between  the  cost  and  the  selling  price 
of  an  article?     Give  the  four  formulas  in  profit  and  loss. 

41.  What  is  meant  by  commission?  What  is  the  difference  between 
a  factor  and  a  broker  ?  Give  the  four  formulas  in  commission  and 
brokerage. 

42.  What  is  the  difference  between  the  market  value  and  the  par 
value  of  capital?  When  is  capital  at  a  premium?  When  is  it  at  a 
discount? 

43.  What  is  the  difference  between  a  dividend  and  an  assessment? 
How  is  the  rate  of  dividend  found  ? 

44.  What  is  insurance  ?  What  is  fire  insurance  ?  What  is  the 
premium?     Give  the  formulas  covering  the  four  cases  in  insurance. 

45.  What  is  life  insurance  ?  How  is  the  premium  computed  ? 
What  is  a  mutual  insurance  company  ? 

46.  What  is  the  difference  between  a  poll  tax  and  a  property  tax? 
How  is  a  property  tax  assessed  ?     How  is  the  rate  of  tax  determined  ? 

47.  What  is  an  income  tax?  An  excise  tax?  From  what  kind  of 
taxes  is  the  internal  revenue  of  the  United  States  derived  ? 

48.  What  are  customs  or  duties?  What  is  the  difference  between 
specific,  duties  and  ad  valorem  duties  ?     What  is  a  tariff? 

49.  What  is  interest?     What  is  the  rate  of  interest? 

50.  How  is  the  interest  of  any  principal  for  one  year,  at  any  rate 
per  cent.,  found  ?  Give  the  formula  for  the  general  method  of  com- 
puting interest.     Give  the  formula  for  the  six  per  cent,  method. 

51.  How  many  methods  are  there  of  finding  the  time  between  two 
dates?     Which  is  called  the  method  by  days? 

52.  On  what  principle  is  the  United  States  Rule  for  partial  pay- 
ments based?    What  rule  is  used  when  a  note  runs  less  than  a  year? 

53.  What  quantities  are  considered  in  interest?  State  the  five 
problems  in  interest,  and  give  the  formula  for  each. 


TEST  QUESTIONS.  285 

54.  What  is  discount?  What  is  the  difference  between  true  dis- 
count and  interest?  Between  true  discount  and  bank  discount?  Be- 
tween bank  discount  and  interest? 

55.  What  is  meant  by  days  of  grace?  When  does  a  note  with 
grace  become  due?  How  is  a  note  not  drawing  interest  discounted 
by  a  bank  ?     How  is  a  note  drawing  interest  discounted  ? 

56.  What  is  a  promissory  note  ?  What  is  its  face  ?  Who  is  an  in- 
dorser?     When  is  a  note  negotiable?    When  is  a  note  not  negotiable? 

57.  What  is  a  draft  ?  What  are  the  names  of  the  three  parties 
named  in  a  draft  ?  What  is  meant  by  the  acceptance  of  a  draft  ?  By 
its  protest? 

58.  What  is  a  bond  ?  What  is  a  coupon  ?  When  bonds  are  quoted 
at  108,  what  are  they  worth?  Name  the  three  principal  classes  of 
United  States  Bonds. 

59.  What  is  annual  interest  ?  When  annual  interest  is  not  paid 
when  due,  what  kind  of  interest  does  it  draw  until  paid? 

60.  What  is  compound  interest  ?  In  what  respect  does  compound 
interest  differ  from  annual  interest  ? 

61.  On  what  principle  is  the  common  method  of  finding  the  equated 
time  of  several  debts  or  payments  based  ?  What  is  meant  by  the  equa- 
tion of  accounts  ? 

62.  Define  ratio.  In  how  many  and  what  ways  may  the  ratio  of 
two  numbers  be  expressed  ?  What  are  the  two  terms  of  a  ratio 
called  ?  Which  is  the  dividend  ?  When  is  the  value  of  a  ratio  less 
than  one?     When  is  it  greater  than  one? 

G3.  Why  must  the  two  terms  of  a  ratio  be  like  numbers?  Why  is 
the  value  of  a  ratio  not  changed  by  multiplying  or  dividing  both  of 
its  terms  by  the  same  numbers? 

64.  What  is  a  compound  ratio?  How  is  a  compound  ratio  reduced 
to  a  simple  ratio? 

65.  What  is  a  proportion  ?  How  many  ratios  in  a  simple  propor- 
tion ?  When  is  a  proportion  called  simple?  When  is  it  comjDound? 
How  many  terras  in  a  simple  proportion  ? 

66.  Which  terms  are  called  the  extremes,  and  which  the  means? 
To  what  is  the  product  of  the  extremes  equal? 

67.  How  can  a  missing  mean  be  found  ?  Why  ?  A  missing  ex- 
treme? Why?  If  the  second  term  of  a  proportion  is  greater  than 
the  first  term,  how  will  the  fourth  term  compare  with  the  third  ? 

68.  In  stating  a  problem  in  proportion,  which  number  is  made  the 
third  term  ?  Why  ?  What  is  the  relation  between  the  ratio  of  like 
causes  and  the  ratio  of  their  effects  ? 

69.  How  may  a  compound  proportion  be  reduced  to  a  simple  propor- 
tion?   How  may  the  fourth  term  of  a  compound  proportion  be  found? 


286  COMPLETE  ARITHMETIC. 

70.  What  is  the  difference  between  a  simple  partnership  and  a  com- 
pound partnership  ?  On  what  does  the  partnership  value  of  capital 
depend  ? 

71.  What  is  the  difference  between  the  power  of  a  number  and  its 
root?  Give  examples.  What  is  the  difference  between  involution 
and  evolution? 

72.  What  is  the  difference  between  a  perfect  power  and  an  imper- 
fect power?     Give  examples.     When  is  a  root  called  a  sm^d  f 

73.  To  what  is  the  square  of  a  number  composed  of  tens  and  units 
equal  ?  To  what  is  the  cube  of  a  number  composed  of  tens  and  units 
equal  ? 

74.  How  many  orders  in  the  square  of  any  number?  How  many 
orders  in  the  square  root  of  any  number?  How  many  orders  in  the 
cube  of  any  number?  How  many  orders  in  the  cube  root  of  any 
number  ? 

75.  How  is  the  first  term  of  the  square  root  of  any  number  found  ? 
The  second  term?  How  is  the  first  term  of  the  cube  root  of  any 
number  found?    The  second  term  ? 

76.  To  what  is  the  square  of  the  hypotenuse  of  a  right-angled  tri- 
angle equal?     The  square  of  the  base  or  perpendicular? 

77.  How  may  the  area  of  a  circle  be  found  ?  When  the  area  is 
given,  how  may  the  diameter  be  found?  What  is  the  relation  be- 
tween the  areas  of  two  circles? 

78.  How  is  the  surface  of  a  sphere  found?  Its  solidity?  What  is 
the  relation  between  the  solid  contents  of  two  spheres? 


APPENDIX. 


NOTATION. 

423.  In  the  decimal  system  of  notation,  with  ten  for  its 
base,  ten  figures  are  used ;  in  a  system  with  twenty  for  its 
base,  twenty  figures  would  be  needed ;  in  a  system  with  five 
for  its  base,  only  five  figures  (1,  2,  S,%  0)  would  be  needed; 
and,  generally,  a  system  of  notation  requires  as  many  different 
figures  as  there  are  units  in  its  base. 

424.  In  a  system  with  five  for  its  base,  24  would  express 
fourteen;  124  would  express  thirty-nine;  1120  would  express 
one  hundred  and  sixty. 

EXERCISES. 

1.  What  number  is  expressed  by  200  on  a  scale  of  five? 

2.  What  number  is  expressed  by  1240  on  a  scale  of  five? 

3.  Express  forty  on  a  scale  of  five. 

4.  Express  one  hundred  on  a  scale  of  five. 

5.  Express  two  hundred  on  a  scale  of  five. 


PKOOF  OF  THE  SIMPLE  RULES  BY  -  CASTING 
OUT  THE  9's." 

425.  The  method  of  proving  the  elementary  operations 
of  arithmetic  by  "casting  out  the  9's"  is  based  on  the 
principle,  that  the  excess  of  9's  m  any  number  is  equal  to  the 
excess  of  9's  in  the  sum  of  its  digits. 

Take,  for  example,  2345.  Dividing  it  by  9,  we  have  the 
remainder  5,  for  the  excess  of  9's ;  and  adding  the  digits 
(2-{-S-\-4-{-5  =  14),  and  dividing  the  sum  by  9,  we  have 
the  same  remainder. 

(287) 


288  COMPLETE  ARITHMETIC. 

426.  This  principle  may  be  thus  explained : 

2000  =:  222  X  9  +  2 

300  =    33  X  9  -h  3 

40  =      4x9  +  4 

5=  5 


2345 


It  is  seen  that  2000  is  222  times  9,  with  a  remainder  2 ; 
300  is  33  times  9,  with  a  remainder  3 ;  40  is  4  times  9,  with 
a  remainder  4.  Hence,  the  remainders  obtained  by  dividing 
the  several  parts  of  a  number,  denoted  by  the  local  value 
of  its  digits,  by  9,  are  respectively  the  digits  of  the  number; 
and  the  remainder  obtained  by  dividing  the  number  itself 
by  9,  equals  the  remainder  obtained  by  dividing  the  sum 
of  its  digits  by  9.     Hence, 

The  excess  of  9's  in  any  number  is  found  by  adding  its  digits 
and  finding  the  excess  of  9's  in  their  sum. 

427.   Proof  of  Addition. 

Process.  The  excess  of  9's  in  the  first  number,  found  by 

one    rr  1         adding  its  digits,  is  1 ;  in  the  second  number,  4 ;   in 

25(5  <'  4  the  third,  7.  The  excess  of  9's  in  the  sum  of  these 
358  "  7  excesses  is  3,  which  equals  the  excess  of  9's  in  939, 
939       "       3        the  amount.     Hence, 

The  excess  of  9^s  in  the  sum  of  several  numbers  is 
equal  to  the  excess  of  9's  in  the  sum  of  their  excesses. 

1.  Add  and  prove  2346,  5084,  6784,  8653,  and  9045. 

2.  Add  and  prove  30483,  50678,  346864,  and  706037. 

3.  Add  and  prove  530902,  672084,  567084,  and  1345602. 


Process. 

3676  Excess  4 

1508       "      5 

2168       "      8 

"      4 


428.   Proof  of  Subtraction. 

Since  the  minuend  is  equal  to  the  sum  of  the  sub- 
trahend and  remainder,  the  excess  of  9's  in  the  minuend 
equals  the  excess  of  9'.s  in  the  SUM  of  the  excesses  in  the 
subtrahend  and  remainder. 


1.  From  40603  take  27475,  and  prove  the  result. 

2.  From  607853  take  492097,  and  prove  the  result. 


CIRCULATING  DECIMALS. 


289 


429.   Proof  of  Multiplication. 


Process. 


347  Excess    5 

58       "       _8 

1041  40 

1735^ 
18391  Excess    4 


Since  347  contains  a  certain  number  of  9's  with 
an  excess  of  5,  and  53  contains  a  certain  number 
of  9's  with  an  excess  of  8,  the  product  of  347  and 
53  consists  of  the  product  of  the  number  of  9's 
in  them,  plus  the  product  of  5  and  8,  the  excesses 
of  9's.     Hence, 

IVie  excess  of  9's  in  the.  product  of  two  numbers  is 
equal  to  the  excess  of  9's  in  the  product  of  the  excesses  in  these  numbers. 

1.  Multiply  45603  by  708,  and  prove  the  result. 

2.  Multiply  60875  by  690,  and  prove  the  result. 

430.  Proof  of  Division. 

Process. 

347  )  18496  (  53 
1735^ 
1146 
1041 


105 


18496  Excess  1 

347      "       5 

53      "       8 

105       "       6 

5  X  8  -f  6  ==:  46       "       1 


Since  the  dividend  equals  the  product 
of  divisor  and  quotient,  plus  the  re- 
mainder, the  excess  of  9's  in  the  divi- 
dend is  equal  to  the  excess  of  9's  in  the 
product  of  divisor  and  quotient,  plus  the 
excess  in  the  remainder.     Hence, 

The  excess  of  9'.s'  in  the  dividend  is  equcd 
to  the  excess  of  9's  in  the  product  of  the 
excesses  in  divisor  and  quotient,  plus  the 
excess  hi  the  remainder. 


1.  Divide  6480  by  47,  and  prove  the  result. 

2.  Divide  15685  by  625,  and  prove  the  result. 


CIRCULATING  DECIMALS. 

431.  A  Circulating  Decimal  is  an  interminate 
decimal,  containing  the  same  figure  or  set  of  figures, 
repeated  in  the  same  order  indefinitely.     (Art.  12L) 

432.  The  figure  or  set  of  figures  repeated  is  called  a 
Bepetend. 

A  repetend  is  denoted  by  a  dot  placed  over  the  first  and 
last  of  its  figures;  as,  .5  .16   .325. 

f.Ar.-i'). 


290  COMPLETE  ARITHMETIC. 

433.  When  a  circulating  decimal  has  no  figure  but  the 
repetend,  it  is  called  a  Pure  Circulate;  as,  .325. 

When  a  circulating  decimal  has  one  or  more  figures 
before  the  repetend,  it  is  called  a  Mixed  Circulate;  as, 
.4526. 

434.  A  pure  circulate  is  reduced  to  a  common  fraction  by 
taking  the  repetend  for  the  numerator,  and  as  many  9's  for  the 
denominator  as  there  are  figures  in  the  repetend. 

Proof. 
Let  .63  be  a  pure  circulate. 

Then,  63.63  =^100  times  the  pure  circulate. 

■63  =      1  time     "       "  " 

Subtracting,     63.       ^    99  times   "       "  " 

Hence,  —        —  the  value  of       '^  " 

99 

435.  A  mixed  circulate  is  reduced  to  a  common  fraction 
by  subtracting  the  terms  which  precede  the  repetend  from  the  ivhole 
repetend,  and  taking  the  difference  for  the  numerator;  and,  for 
the  denominator,  taking  as  many  9's  as  there  are  figures  in  the 
repetend,  with  as  many  ciphers  annexed  as  there  are  decimal 
figures  before  the  repetend. 

Proof. 
Let  .45124  be  a  mixed  circulate. 

Then,  45124.124  =  100000  times  the  mixed  circulate. 

And,  45.124  =        100     "      "      "  " 

Subtracting,  45079         =    99900     "       "       " 

Hence,  fffj^         =  the  value  of     "      "  " 

436.  Pure  or  mixed  circulates  may  be  added,  subtracted, 
multiplied,  or  divided  by  first  reducing  them  to  common 
fractions. 

Note. — Circulates  may  be  added,  subtracted,  multiplied,  or  divided 
without  tirst  reducing  them  to  common  fractions;  but  the  processes  are 
not  of  sufficient  practical  importance  to  justify  their  explanation  in  a 
school  arithmetic.  In  all  computations,  circulates  are  carried  to  enough 
places  to  avoid  any  appreciable  error  in  the  result,  and  then  are  treated 
as  other  decimals. 


TABLES. 


291 


437.  TABLES  OF  DENOMINATE  NUMBERS. 


I.  CURRENCIES. 


1.  United  States  Money. 

The  denominations  are 
mills,  cents,  dimes,  dollars, 
and    ec 


Table. 

10  rn.  =    1  ct. 
10  ct.  =    I  d. 
10  d.   =  $1 
$10       =    \  E. 


2.  English  Money. 

The  denominations  are/ar- 
tkings  (q.),  j)ence  (d.),  shil- 
lings (s.),  and  pounds  (£). 

Table. 

4q.=--ld. 

Ud.  =  l  s. 

20  s.  =  1  £. 

1  £  =  $4.84. 


II.  MEASURES  OF  EXTENSION  AND  TIME. 


1.    MEASUKES  OF   LINES  AND  AKCS. 


Long  Measure. 

The  denominations  are 
inches,  feet,  yards,  rods,  fur- 
longs, and  miles. 

Table. 


12  in. 

^\ft. 

3  A 

=  lyd. 

5|  yd. 

=  1  rd. 

40  rd. 

=  1  /wr. 

8  fur. 

=  1  m. 

Also: 

3  barleycorn. 

,•  =  1  incA. 

4  mcAes 

=  1  hand. 

3/ee< 

■=  1  pace. 

6  feet 

=  1  fathom. 

3  wuVes 

^rr  1  league. 

60  geographic 

miles           -y 
les  {nearly)  f 

69 1  statute  nu 

Circular  Measure. 

The  denominations  are  sec- 
onds, minutes,  degrees,  signs, 
and  circumferences. 

Table. 
60^^      =  y 
60'     =  r 

30°       -=  1  s. 
12  s. 
360° 


}  = 


IC. 


Cloth  Measure. 

(Little  used.) 

2\in.^=  1  nxiil. 

4  w.     =1  quarter. 
A.  qr.   =  1  yard. 

5  gr.   =  1  Ell  Eng. 


1  degree  at  the  equator. 


292 


COMPLETE  ARITHMETIC. 


2.  MEASURES  OF  SURFACES  OR  AREAS. 


Square  Measure. 

The  denominations  are 
square  inches,  square  feet, 
square  yards,  square  rods  (or 
perches),  roods,  acres,  and 
square  miles. 

Table. 


144  sq. 

in. 

=  1  sq. 

ft. 

9  sq. 

ft' 

^Isq. 

yd. 

SO\sq 

.yd. 

=  1  P. 

40  p. 

=  li2. 

4i2. 

=  1  A. 

640  A. 

=:  1  sq. 

mi. 

Surveyor's  Measure. 


Table. 

7.92  in. 

=  1  link  (/.). 

2b  I. 

=  1  rod. 

4  rd. 

=  1  chain  (ch.) 

80  ch. 

=  1  mile. 

Also: 

625  sq.  I 

.      -=1  P. 

16  P. 

=  1  so.  ch. 

10  sq.  ch.  =1  A. 

640  A. 

■=  1  sq.  mi. 

1  sq.  mi.  =^  1  section. 

36  sect. 

=  1  township 

3.  MEASURES  OF  SOLID  CONTENTS  OR  CAPACITY. 


Cubic  Measure. 

The  denominations  are 
cubic  inches,  cubic  feet,  and 
cubic  yards. 

Table. 
1728  CM.  in.  =  1  cw.  ft. 
27  cu.  ft.  =  1  cu.  yd. 

Wood  Measure. 

Table. 
16  cu.  ft.  =  1  cord  ft. 

^'^'f^-^n  =  lcord. 
128  cu.  ft.       / 

24|  cu.ft.  =  1  perch  of  stone. 
40  cu.  ft.  round  timber  ==  1  ton. 
50  cu.  ft.  hewn  timber  =  1  ton. 

Note. — The  standard  bushel 
gallon,  231  cu.  in. ;  and  tlie  beer 


Dry  Measure. 

The  denominations  are 
pints,  quarts,  pecks,  and 
busJiels. 

Table. 
2  pt.  =1  qt. 


8  qt. 

4pk. 


Ipk. 
1  bu. 


Liquid  Measure. 

Table. 

4  gills   =  1  pt. 

2pt.      =1  qt. 

4  qt.      =1  gal. 
3U  gal.  =--  1  66;. 
63  gal.    =  1  hhd. 
42  gal.    =  1  tierce. 

contains  2150|  cu.  in. ;   the  liquid 
gallon  (little  used),  282  cu.  in. 


TABLES. 


293 


4.  measures  of  duration  or  time. 
Time  Measure. 

The  denominations  are  seconds,  minutes,  hours,  days,  years, 
and  centuries. 


Table. 

60  sec.    -—  1  min. 
60  min.  --  1  A. 
24  A.      =1  da. 

365  da.    :=  1  common  yr. 

366  da.  ^^  1  leap  yr. 
365^  da.  =  1  sola7'  yr. 
100  s.  ?/7'.  =;  1  century. 

Also  : 

7  da.  ^-^  1  week. 
4  w.   :^  I  lunar  mo. 


Calendar  Months. 


January, 

1st  mo. 

31  days. 

February, 

2d 

u 

28  or  29 

March, 

3d 

u 

81  days. 

April, 

4th 

" 

30      " 

May, 

5th 

il 

31      " 

June, 

6th 

ii 

80      '' 

July, 

7th 

il 

31      " 

August, 

8th 

il 

31      " 

September, 

9th 

ii 

80      " 

October, 

10th 

,1 

81      '' 

November, 

11th 

ii 

30      " 

December, 

12th 

11 

31      " 

Also  : 

A  Julian  year  contains  13  lunar  mo.  1  da.  6  h. 

A  civil  year  contains  12  calendar  months. 

A  solar  year  contains  365  da.  5  h.  48  min.  48  sec. 


III.  WEIGHTS. 
Avoirdupois  Weight. 

The  denominations  are  drams,    ounces,   pounds,   hundred- 
weights, and  tons. 

Table. 


16  dr. 

r=loz. 

16  oz. 

=  1/6. 

100  lb. 

=^  1  cwt. 

20  cwt 

.  -  1  T. 

Ako  : 

196  lb. 

flour 

=  1 

200  lb. 

beef  or 

pork    =  1 

barrel. 


100  lb.  fish 
56  lb.  corn  or  rye  ^ 
60  lb.  tvheat  > 

32  lb.  outs  J 

14  lb.  iron  or  lead 
2H  stones 
8  pigs 


=  1   quintal. 

=^  1  bushel. 

■=  1  stone. 
=  1  pig. 
=  1  /other. 


294 


COMPLETE  ARITHMETIC. 


Troy  Weight. 

The  denominations  are 
grains,  'pennyweights,  ounces, 
and  pounds. 

Table. 

24  gr.    =  1  pivt. 
20  pwt.  =  1  oz. 
12  oz.    =1  lb. 

4  ^T.  =  1  carat. 


Apothecaries  Weight. 

The  denominations  are 
grains,  scruples,  drams, 
ounces,    and    pounds. 

Table. 
20  gr.  =  16 

36=13 

83=13 
12  3    =  1  tb 


COMPARISON  OF  WEIGHTS. 
1  ^6.  Avoir.  =  IxVi  lb.  Troy  =  1  yVf  ^  ^poth. 

1  oz.    "    =  n%  oz.  "  =  lii  3 


IV.  MISCELLANEOUS  TABLE. 


12  thirigs 

are 

1 

dozen. 

12    (^02671 

(( 

1 

gi'oss. 

12  gross 

« 

1 

great  gross. 

20  fAiw^rs 

u 

1 

score. 

18  iTicAes 

a 

1 

cubit. 

22  inches  (nearly)    " 

1 

sacred  cubit. 

Paper 

. 

24  sheets 

are 

1 

quire. 

20  ^mVes 

a 

1 

ream. 

2  reams 

a 

1 

bundle. 

5  bundles 

n 

1 

bale. 

Books 

. 

sheet  folded  in  2  leaves  - 

's  called 

a  folio. 

<< 

4 

a  qwirto,  or 

4/0. 

« 

8 

an  octa.vo,  or 

8ro. 

« 

12 

a  duodecimo, 

or  12mo. 

« 

16 

a  16wio. 

« 

24 

a  24wo. 

Note. — In  estimating  the  size  of  the  leaves,  as  above,  the  double 
medium  sheet  (23  by  26  inches)  is  taken  as  a  standard. 


LIFE  INSURANCE.  295 

LEGAL  RATES  OF  INTEREST  IN  THE  SEVERAL 
STATES. 

438.  When  no  rate  is  mentioned,  the  legal  rate  in 

Louisiana,  except  on  bank  interest  (6%),  is 5  % 

New  York,  New  Jersey,  Michigan,  Wisconsin,  Minnesota,  South 

Carolina,  and  Georgia 7  % 

Alabama  and  Texas 8  % 

California,  Oregon,  Kansas,  Nebraska,  Nevada,  and  Colorado    .  10  fo 
All  the  other  States  and  District  of  Columbia 6  % 

When  stipulated  in  the  contract,  the  legal  rate  in 

Ohio,  Florida,  and  Louisiana  is  as  high  as 8  % 

Illinois,  Iowa,  Michigan,  Arkansas,  Mississippi,  Missouri,  and 

Tennessee 10  % 

Minnesota,  Texas,  Wisconsin 12  % 

Nebraska 15  % 

Kansas 20  % 

Massachusetts,  Rhode  Island,  California,  Nevada,  and  Colorado, 

any  per  cent,  agreed  upon. 

The  legal  rate  in  England  and  France  is 5  % 

Canada,  Nova  Scotia,  and  Ireland 0  % 

Note. — Since  the  rate  of  interest  is  often  changed  by  legislation,  the 
above  rates  may  not  in  all  cases  be  strictly  accurate. 

LIFE   INSURANCE. 

439.  The  rate  of  premium  in  life  insurance  is  based  on 
the  applicant's  expectation  of  life,  as  shown  by  life  statistics 
or  bills  of  mortality. 

The  annual  premium  must  be  such  a  sum  as,  when  put 
at  interest,  will  amount  to  the  sum  insured  at  the  close  of 
average  extension  of  life  beyond  the  applicant's  age. 

440.  There  are  two  tables  showing  the  Expectation  of 
Life,  called  the  Carlisle  Table  and  the  Wigglesworth  Table. 
The  former  is  based  on  bills  of  mortality  prepared  in  Eng- 
land, and  the  latter  is  based  on  the  mortality  in  the  United 
States.     Both  tables  are  in  use  in  this  country. 


296 


COMPLETE  AUITIIMETIC. 


441.  The    Expectation    of  Life,    as    shown    by   the    two 
tables,  is  as  follows  : 


n 

> 

a 

> 

n 

>■ 

n 

H 

a 

1 

>< 

w 
o 
< 

Is 

X 

-< 

5w 

o 
-< 

0 

38.72 

28.15 

24 

38.59 

32.70 

48 

22.80 

22.27 

72 

8.16 

1 

44.68 

36.78 

25 

37.86 

32.33 

49 

21.81 

21.72 

73 

7.72 

2 

47.55 

38.74 

26 

37.14 

31.93 

50 

21.11 

21.17 

74 

7.33 

3 

49.82 

40.01 

27 

36.41 

31.50 

51 

20.39 

20.61 

75 

7.61 

4 

50.76 

40.73 

28 

35.69 

31.08 

52 

19.68 

20.05 

76 

6.49 

5 

51.25 

40.88 

29 

35.00 

30.66 

53 

18.97 

19.49 

77 

6.10 

6 

51.17 

40.69 

30 

34.34 

30.25 

54 

18.28 

18.92 

78 

6.02! 

7 

50.80 

40.47 

31 

33.68 

29.83 

55 

17.58 

18.35 

79 

5.80 

8 

50.24 

40.14 

32 

33.03 

29.43 

56 

16.89 

17.78 

80 

5.51 

9 

49.57 

39.72 

33 

32.36 

29.02 

57 

16.21 

17.20 

81 

5.21 

10 

48.82 

39.23 

34 

31.68 

28.62 

58 

15.55 

16.63 

82 

4.93 

11 

48.04 

38.64 

35 

31.00 

28.22 

59 

14.92 

19.04 

83 

4.65 

12 

47.27 

38.02 

36 

30.32 

27.78 

60 

14.34 

15.45 

84 

4.39 

13 

46.51 

37.41 

37 

29.64 

27.34 

61 

13.82 

14.86 

85 

4.12 

14 

45.75 

36.79 

38 

28.96 

26.91 

62 

13.31 

14.26 

86 

3.90 

15 

45.00 

36.17 

39 

28.28 

26.47 

63 

12.81 

13.66 

87 

3.71 

16 

44.27 

35.76 

40 

27.61 

26.04 

64 

12.30 

13.05 

88 

3.59 

17 

43.57 

35.37 

41 

26.97 

25.61 

65 

11.79 

12.43 

89 

3.47 

18 

42.87 

34.98 

42 

26.34 

25.19 

66 

11.27 

11.96 

90 

3.28 

19 

42.17 

34  59 

43 

25.71 

24.77 

67 

10.75 

11.48 

91 

3  26 

20 

41.46 

34.22 

44 

25.09 

24.35 

68 

10.23 

11.01 

92 

3.37 

21 

40.75 

33.84 

45 

24.46 

23.92 

69 

9.70 

10  50 

93 

3.48 

22 

40.04 

33.46 

46 

23.82 

23.37 

70 

9.18 

10.06 

94 

3.53 

23 

39.31 

33.08 

47 

2317 

22.83 

71 

8.65 

9.60 

95 

3.53 

X 

9.14 

8.69 
8.25 
7.83 
7.40 
6.99 
6.59 
6.21 
5.85 
5.50 
5.16 
4.87 
4.66 
4.57 
4.21 
3.90 
3.67 
3.56 
3.73 
3.32 
3.12 
2.40 
1.98 
1.62 


Note. — A  comparison  shows  that  the  Wij?glesworth  table  has  a 
less  expectation  of  life  than  tlie  Carlisle  table  for  all  ages  below  50 
years ;  and  that  the  latter  table  has  a  less  expectation  than  the  former 
for  all  ages  from  50  to  90  years  inchisive. 


EQUATION   OF  PAYMENTS. 

442.  In  1860,  the  author  published  a  demonstration  of  the 
correctness  of  the  common  INIercantile  Kule  for  finding  the 
equated  ti-me  for  the  payment  of  several  debts,  due  at  dif- 
ferent times  without  interest.  The  inaccuracy  of  the  rule 
hy  present  ivorths,  commended  by  several  authors  as  '^the  only 
accurate  ride'^yvas  thus  pointed  out: 


ARITHMETICAL  PROGRESSION.  297 

"  The  equated  time  for  the  payment  of  $200,  of  which  $100  is  now 
due,  and  the  other  $100  is  due  in  two  years,  as  found  by  this  rule,  is 
11.32  months.  Now,  the  amount  of  $100  for  11.32  months,  at  6  per 
cent.,  is  $105.66;  the  present  worth  of  the  other  $100,  due  in  12.679 
months,  is  $94,038,  and  $105.66  +  $94,038  :=  $199,698,  whereas  it 
ought  to  be  $200. 

"  It  is  also  evident  that  the  equated  time,  as  found  by  this  'accurate' 
rule,  will  not  be  the  same  for  all  rates  of  interest.  At  50  per  cent, 
the  equated  time  of  the  above  example  is  8  months,  and  the  error,  by 
the  above  test,  $8.33^;  at  100  per  cent,  it  is  6  months,  with  an  error 
of  $10. 

"This  supposed  accurate  rule  is  based  upon  the  principle  that  the 
amount  to  be  paid  on  a  debt  due  at  a  future  date,  witiiout  interest,  at 
any  time  previous  to  this  date,  is  the  present  worth  of  the  debt  at  any  prior 
date,  plus  the  interest  of  the  present  worth  up  to  date  of  payment. 
The  incorrectness  of  this  principle  is  easily  shown.  Suppose  I  owe  a 
man  $100,  due  in  two  years,  without  interest;  how  much  ought  I  to 
pay  in  one  year? 

"The  present  worth  of  $100,  due  in  tAvo  years  (at  6  per  cent.),  is 
$89.2857,  and  the  interest  on  this  sum  for  one  year  is  $5.3571 ;  hence, 
the  sum  to  be  paid  is  $89.2857  +  $5.3571  =  $94.6428.  The  true 
amount  to  be  paid,  however,  is  the  present  worth  of  $100,  due  in  one 
year,  which  is  $94,339." 

Note. — The  accuracy  of  the  Mercantile  Rule  and  the  inaccuracy 
of  the  rule  by  Present  Worths  were  rigidly  demonstrated  by  Prof.  A. 
Schuyler,  in  an  article  published  in  the  Ohio  Educational  JHonthly,  for 
1862,  p.  116. 


ARITHMETICAL  PROGRESSIOK 

443.  An  A  ritlimetical  JProgression  is  a  series  of 
numbers  which  so  increases  or  decreases  that  the  diiference 
between  the  consecutive  numbers  is  constant. 

444.  The  numbers  which  form  the  series  are  called 
Terms,  the  first  and  last  terms  being  the  ExtremeSy  and  the 
intervening  terms  the  Means. 

The  difference  between  the  consecutive  terms  is  called  the 
Common  Difference. 

445.  An  Ascending  Series  is  one  in  which  the  terms  in- 
crease; as,  2,  5,  8,  11,  14,  etc. 

A  Descending  Seines  is  on€  in  which  the  terms  decrease ; 
as  20,  17,  14,  11,  8,  etc. 

446.  In  an   arithmetical  progression   five  quantities   are 


298  COMPLETE  ARITHMETIC. 

considered ;  and  such  is  the  relation  between  them,  that,  if 
any  three  are  given,  the  other  two  may  be  found. 

These  quantities  are: 

1.  The  first  term. 

2.  The  last  term. 

3.  The  common  difference. 

4.  TJie  number  of  terms. 

5.  The  sum  of  all  the  terms. 

447.  The  ascending  series,  2,  5,  8,  11,  14,  having  5  terms, 
may  be  expressed  in  three  forms,  as  follows : 


(1) 

2 

5 

8 

11 

14 

(2) 

2 

2+3 

2+(3+3) 

2+(3+3+3) 

2+(3+3+3+3) 

(3) 

2 

2+3 

2+3X2 

2+3X3 

2+3X4 

A  comparison  of  these  three  forms  of  the  same  series 
shows,  that  each  term  is  composed  of  two  parts,  viz.:  (1)  the 
first  term ;  (2)  the  common  difference  taken  as  many  times 
as  there  are  preceding  terms.     Hence, 

1.  TJie  last  term  of  an  ascending  series  is  equal  to  the  first 
term,  plus  the  common  difference  taken  as  many  times  as  there 
are  terms  in  the  series  less  one.     Conversely, 

2.  The  first  term  of  an  ascending  series  is  equal  to  the  last 
term,  minus  the  common  difference  taken  as  many  times  as  there 
are  terms  in  the  series  less  one. 

3.  The  common  difference  is  equal  to  the  difference  between 
the  first  and  last  terms,  divided  by  the  jiumber  of  terms  less  one. 

4.  TJw  number  of  terms  less  one  is  equal  to  the  difference 
between  the  first  and  last  terms,  divided  by  the  common  difference. 

448.  Let 

3        5        7        9       11       13  be  an  arithmetical  series, 
and,       13       11         9        7        5        3  be  the  series  reversed. 

Then,    16  +  16  +  16  +  16  +  16  +  16  =  twice  the  sum  of  the  terms. 

and  8  -f    8  +    8  +    8  +    8  +    8  =  the  sum  of  the  terms. 


ARITHMETICAL  PROGRESSION.  299 

An  inspection  of  the  above  shows  that  the  sum  of  the  first 
and  last  terms  of  an  arithmetical  series,  multiplied  by  the 
number  of  terms,  is  equal  to  twice  the  sum  of  all  the  terms. 
Hence,  The  sum  of  all  Hie  terms  of  an  arithmetical  series  is 
equal  to  the  product  of  one  half  the  sum  of  the  first  and  last 
terms,  multiplied  by  the  number  of  terms. 

Note. — One  half  of  the  sum  of  the  first  and  last  terms  is  equal  to 
the  average  of  the  several  terms  of  the  series. 

449.  From  the  above  principles  may  be  deduced  the  fol- 
lowing 

FORMULAS. 

1.  Last  term  =  first  term  dz  (com.  difference  X  number  of 
terms  less  one). 

2.  First  term  =  last  term  =h  (com.  difference  X  number  of 
term^  less  one). 

3.  Common  differeme  =  |  '«**  '^'■™  -fi'"^  '^'"  I  ^  number 

L  first  term  —  last  term  ) 
of  terms  less  one. 

4.  Number  of  term^  less  on^^\^\f^~fij^,  ^'^X 
•^  (  first  term  —  last  term  J 

-T-  common  dfference. 

5.  Sum  of  terms  =  ^  (first  term  -\-  last  term)  X  number  of 

terms. 

Note. — The  first  term  of  an  ascending  series  corresponds  to  the  last 
term  of  a  like  descending  series,  and  the  last  term  of  a  descending  series 
corresponds  to  the  first  term  of  a  like  ascending  series. 

PBOBLEMS. 

1.  What  is  the  tenth  term  of  the  series  5,  7,  9,  11,  etc.? 

2.  The  first  term  of  an  ascending  series  is  4,  the  common 
difference  3,  and  the  number  of  terms  8 :  what  is  the  last 
term  ? 

3.  The  last  term  of  a  descending  series  is  1,  the  common 
difference  4,  and  the  number  of  terms  12 :  what  is  the  first 
term? 


300  COMPLETE  ARITHMETIC. 

4.  The  extremes  of  an  arithmetical  series  are  47  and  3, 
and  the  number  of  terms  12:  what  is  the  common  differ- 
ence ? 

5.  The  1st  term  is  7  and  the  21st  term  57:  what  is  the 
common  difference? 

6.  The  4th  term  of  a  series  is  21  and  the  9th  term  is  41 : 
what  are  the  four  mean  terms? 

7.  The  two  extremes  of  a  series  are  12  and  177,  and  the 
common  difference  5  :   what  is  the  number  of  terms? 

8.  The  two  extremes  of  a  series  are  20  and  152,  and  the 
number  of  terms  45 :   what  is  the  sum  of  all  the  terms  ? 

9.  What  is  the  sum  of  all  the  terms  of  the  series  described 
in  the  6th  problem  above?     In  the  7th  problem? 

10.  How  many  strokes  does  the  hammer  of  a  clock  make 
in  24  hours? 

11.  A  man  agreed  to  dig  a  trench  50  yards  long  for  2 
cents  for  the  first  yard,  5  cents  for  the  second  yard,  8  cents 
for  the  third,  and  so  on,  the  price  of  each  yard  being  3 
cents  more  than  that  of  the  preceding  yard :  what  did  he 
receive  for  digging  the  last  yard?     For  digging  the  trench? 

GEOMETRICAL  PROGRESSION. 

450.  A  Geometrical  Profp^ession  is  a  series  of 
numbers  which  so  increases  or  decreases  that  the  ratio  be- 
tween the  consecutive  terras  is  constant. 

The  first  and  last  terms  are  called  the  Extremes,  and  the 
intervening  terms  are' called  the  Means. 

451.  A  geometrical  progression  is  ctscending  or  descending 
according  as  the  series  increases  or  decreases  from  left  to 
right. 

452.  In  a  geometrical  progression  five  quantities  are  con- 
sidered, and  these  (as  in  arithmetical  progression)  are  so 
related  to  each  other  that,  any  three  being  given,  the  other 
two  may  be  found. 


GEOMETRICAL  PROGRESSION.  301 

These  five  quantities  are 

1.  The  first  term. 

2.  The  last  term. 

3.  The  common  ratio. 

4.  The  number  of  terms. 

5.  The  sum  of  all  the  terms. 

453.  The  ascending  series,  2,  6,  18,  54,  162,  486,  has  6 
terms,  and  the  first  term  is  2,  and  the  common  ratio  or 
multiplier  is  3.  This  series  may  be  expressed  in  three 
forms,  as  follows: 

(1)  2      6  18  54  162  486 

(2)  2  2X3  2X3X3  2X3X3X3  2X3X3X3X3  2X3X3X3X3X3 

(3)  2  2X3  2X32    2X3^      2X3*        2X3^ 

A  comparison  of  the  corresponding  terms  of  the  three 
forms,  shows  that  each  term  of  the  series  is  composed  of 
two  factors,  viz. :  (1)  the  first  term,  and  (2)  the  common 
ratio  raised  to  a  power  whose  exponent  or  degree  is  equal  to 
the  number  of  preceding  terms.     Hence, 

1.  The  last  term  of  a  geometrical  series  is  equal  to  the  first 
term,  multiplied  by  the  common  ratio,  raised  to  a  power  whose 
degree  is  one  less  than  the  number  of  terms.     Conversely, 

2.  The  first  term  is  equal  to  the  last  term  divided  by  the  com- 
mon ratio,  raised  to  a  power  whose  degree  is  one  less  than  the 
number  of  terms. 

3.  The  common  ratio  is  equal  to  the  root  wh-ose  index  is  one 
less  than  the  number  of  terms,  of  the  quotient  of  the  last  terra 
divided  by  Hie  first  term. 

454.  By  an  algebraic  process  it  may  be  shown  that 

4.  The  sum  of  a  geometrical  series  is  equal  to  the  product  of 
the  last  term  and  the  common  ratio,  less  the  first  term,  divided 
by  the  common  ratio  less  one. 

455.  When  the  number  of  terms  in  a  descending  geomet- 
rical series  is  infinite,  the  last  term  is  0,  and  the  sum  of  the 
series  is  equal  to  the  first  term  divided  by  the  ratio  less  one. 


302  COMPLETE  ARITHMETIC. 

456.  From  the  above  principles  may  be  deduced  the  fol- 
lowing 

FORMULAS. 

1.  Last  term  =  first  term  X  ratio^^~^. 

2.  First  term  =  last  term  ■—  ratio  "~  ^ . 


3.  Matio  =    -[/last  term  -v- first  term. 

.CI         /.       •         (^^^  ^''"'^  X  ratio)  —  iirst  term 

4.  bum  of  series  = -. — ' 

-^  ratio  —  1 

5.  Sum  of  infinite  descending  series  =  first  term  -~  (ratio — 1 ). 

Notes. — 1.  By  "ratio  "-i,"  in  1st  and  2d  formulas,  is  meant  the 
ratio  raised  to  a  power  whose  degree  is  the  number  of  terms  less  1. 
The  index  of  the  root,  in  the  3d  formula  {n — 1),  is  the  number  of 
terms  less  ]. 

2.  In  an  ascending  series  the  ratio  is  greater  than  1,  and  in  a  de- 
scending series  the  ratio  is  less  than  1. 

PROBLEMS. 

1.  What  is  the  6th  term  of  the  series  5,  10,  20?  etc. 

2.  The  first  term  of  a  geometrical  series  is  5,  the  ratio  is 
3,  and  the  number  of  terms  7 :  what  is  the  last  term  ? 

3.  The  first  term  of  a  series  is  1220,  the  ratio  ^,  and  the 
number  of  terms  6 :  what  is  the  last  term  ? 

4.  The  last  term  of  a  series  is  64,  the  ratio  2,  and  the 
number  of  terms  10  :  what  is  the  first  term? 

5.  What  is  the  sum  of  the  series  described  in  the  4th 
problem  ?     In  the  3d  problem  ? 

6.  The  first  term  of  a  series  is  5,  and  the  sixth  term  is 
1215:  what  is  the  ratio? 

7.  The  first  term  of  a  series  is  10,  the  sixth  term  2430, 
and  the  ratio  3 :  what  is  the  sum  of  the  six  terms  ? 

8.  A  father  gave  his  son  50  cents  on  his  12th  birthday, 
and  agreed  to  double  the  amount  on  each  succeeding  birth- 
day to  and  including  the  21st :  how  much  did  the  son  re- 
ceive on  his  21st  birthday?     How  much  in  all? 

9.  A  man  worked  15  days  on  condition  that  he  should 
receive  1  cent  the  first  day,  5  cents  the  second  day,  and  so 


ALLIGATION.  303 

on,  the  wages  of  each  day  being  5  times  the  wages  of  the 
previous  day :  how  much  did  he  receive  ? 

ALLIGATION. 

457.  Alligation  is  the  process  of  finding  the  average 
value  or  quality  of  a  mixture  composed  of  articles  of  dif- 
ferent values  or  qualities. 

It  is  also  the  process  of  compounding  several  articles  of 
different  values  or  qualities  to  form  a  mixture  of  an  average 
value  or  quality. 

The  first  process  is  called  Alligation  Medial^  and  the  sec- 
ond Alligation  Alternate. 

Note. — The  term  Alligation  is  derived  from  the  Latin  alligare, 
to  bind  or  link.  The  term  is  applied  to  this  process  because  some  of 
the  problems  may  be  solved  by  joining  or  linking  the  numbers. 

Case  I. 

458.  Several  ingredients  of  a  mixture,  and  their  respective 
values  given,  to  find  their  average  value. 

PROBLEMS. 

1.  A  farmer  mixed  25  bushels  of  oats,  at  50  cents  a 
bushel ;  15  bushels  of  rye,  at  80  cents  a  bushel ;  and  30 
bushels  of  corn,  at  70  cents  a  bushel :  Avhat  was  the  value 
of  a  bushel  of  the  mixture  ? 

Process. 

^A*  V  95  —  mo  ^^"^^  *^^  *°*^^  ^^^"^  ^^  *^^  '^^  bushels 

80  X  15  ^  1200  ^^  grain  mixed  together  was  4550  cents, 

70  X  30  =  2100  the  value  of  1  bushel  was  yV  of  4550  cents. 

70    )  4550  which  is  65  cents. 
65  cts.,  Ans. 

2.  A  grocer  mixed  20  pounds  of  cofiee  worth  28  cents,  30 
pounds  worth  35  cents,  and  50  pounds  worth  33  cents :  what 
is  a  pound  of  the  mixture  worth  ? 


304  COMPLETE  ARITHMETIC. 


Case  II. 

459.  The  values  of  several  articles  given,  to  find  in  what  pro- 
portion they  must  be  compounded  to  make  a  mixture  of  a  given 
value. 

3.  A  grocer  has  sugars  worth  16,  18,  and  24  cents  a 
pound:  in  what  proportion  must  they  be  taken  to  make  a 
mixture  worth  20  cents  a  pound  ? 

I.  Solution  by  Analysis. 

On  each  pound  of  sugar  worth  16  cents  taken,  there  is  a  gain  of  4 
cents,  and  on  each  pound  at  24  cents,  there  is  a  loss  of  4  cents.  Hence, 
these  two  kinds  of  sugar  may  be  taken  in  equal  quantities,  or  1 
pound  of  each.  On  each  pound  worth  18  cents  there  is  a  gain  of  2 
cents,  and  hence  2  pounds  of  it  must  be  taken  to  offset  a  loss  of  4 
cents  on  1  pound  at  24  cents.  Hence,  the  simplest  proportionals  are 
1  lb.  at  16  cts.,  2  lb.  at  18  cts.,  and  2  lb.  at  24  cts. 

II.  Another  Solution. 

1  lb.  at  16  cts.  selling  for  20  cts.  gains  4  cts.  \    />    . 

1      "      18        "         "        20    "       "      2  cts.  i  ^^'''' 

1      "      24        "         "         20    "    loses  4  cts.  .  .  4  cts.  loss. 

Taking  two  pounds  each  of  the  first  two  kinds,  the  loss  will  be  12 
cents,  and  by  taking  3  pounds  of  the  third  kind,  the  loss  will  be  12 
cents.  Hence,  the  proportionals  2,  2,  3  make  the  gains  and  losses 
equal. 


III.  Solution  by  Linking. 


20 


16 

n    .      .      .■     .   4 

18—1 

....  4 

24     1 

J  .  4  4-2  =  6 

2 

2 

(3 


Note. — When  only  two  articles  of  different  values  are  given,  they 
can  be  compounded  in  but  one  way  ;  but  when  more  than  two  articles 
are  given,  they  may  be  compounded  in  an  infinite  number  of  tvays. 
They  may  be  combined  two  and  two  in  such  proportions  as  to  make, 
in  each  case,  a  mixture  of  the  required  value,  and  then  these  com- 
pounds may  be  united  in  any  proportions  whatever. 


APPENDIX.  305 

4.  A  merchant  has  teas  worth  $1.25,  $1.40,  $1.60,  and 
$1.75:  how  much  of  each  kind  must  be  taken  to  make  a 
mixture  worth  $1.50? 

Case  III. 

460.  The  values  of  the  several  ingredients  of  a  mixture,  their 
average  value,  and  the  quantity  of  one  or  more  of  ike  ingredients 
given,  to  find  the  respective  quantities  of  the  other  ingredients. 

5.  A  grocer  wishes  to  mix  100  pounds  of  coffee  at  25  cts, 
with  coffees  at  22,  28,  and  30  cts.,  making  a  mixture  worth 
27  cts.  :  how  much  of  each  kind  must  he  take  ? 

Suggestion. — Find  the  proportionals  of  the  ingredients  by  Case  II, 
and  then  muhiply  each  proportional  by  the  quotient  of  100  lbs.  di- 
vided by  the  proportional  for  the  coffee  worth  25  cts. 

6.  A  farmer  wishes  to  mix  60  bushels  of  corn  at  60  cts. , 
with  rye  at  75  cts.,  barley  at  50  cts.,  and  oats  at  40  cts., 
to  make  a  mixture  worth  65  cts.  :  how  many  bushels  each 
of  rye,  barley,  and  oats  must  he  take  ? 

Case  IV. 

461.  The  values  of  the  ingredients,  and  the  quantity  and 
value  of  the  mixture  given,  to  find  the  quantity  of  each  ingre- 
dient. 

7.  How  much  gold  16  carats  fine,  18  carats  fine,  and  22 
carats  fine,  must  be  taken  to  make  12  rings  20  carats  fine, 
and  weighing  4^  pwt.  each  ? 

Suggestion. — Find  the  proportionals  by  Case  II,  and  then  divide 
the  whole  quantity  into  parts  proportional  to  these  proportionals. 

8.  HoAV  much  sugar  worth  15  cts.,  17  cts.,  and  20  cts. 
must  be  taken  to  make  a  mixture  of  200  pounds,  worth 
18  cts.? 

9.  How  much  water  must  be  mixed  with  vinegar,  worth 
60  cts.  a  gallon,  to  make  90  gallons,  worth  50  cts.  a 
gallon  ? 

C.Av.— 2(^ 


306  COMPLETE  ARITHMETIC. 


DUODECIMALS. 

462.  A  Duodechnal  is  a  denominate  number  in 
which  twelve  units  of  any  denomination  make  a  unit  of 
the  next  higher  denomination. 

A  duodecimal  may  be  regarded  as  a  fraction  Avhose  denominator 
is  a  power  of  12 ;  or  a  number  whose  scale  is  12.  The  term  is  de- 
rived from  the  Latin  duodecim,  twelve. 

463.  Duodecimals  are  used  by  artificers  in  measuring 
surfaces  and  solids. 

The  foot  is  divided  into  primes,  marked  ' ;  the  primes  into 
seconds  (")  ;  the  seconds  into  thirds  ('"),  etc.,  as  is  shown  in 
the  following 

Table. 

12  fourtlis  {'''^)  are  V^^ 
12  thirds  "    V 

12  seconds  "    V 

12  primes  "    1  ft. 

The  accents  used  to  mark  the  different  denominations,  are 
called  Indices. 

464.  The  prime  denotes  the  twelfth  of  a  foot ;  the  second, 
ilie  twelfth  of  the  twelfth  of  a  foot,  etc. 

When  a  duodecimal  denotes  the  area  of  a  surface,  the 
foot  is  a  square  foot;  the  prime,  t]ie  twelfth  of  a  square  foot; 
the  second,  the  twelfth  of  a  twelfth  of  a  square  foot,  etc. 

When  a  duodecimal  denotes  the  contents  of  a  solid,  the 
foot  is  a  cubic  foot ;  the  prime,  the  twelfth  of  a  cubic  foot,  etc. 

465.  ADDITION  AND  SUBTRACTION. 

PROBLEMS. 

1.  Add  12  ft.  8'  11'',  16  ft.  10'  9",  and  24  ft.  6". 
Process:  J  i«  ^*  i"'    Q'^ 


f  12  ft.    8"  ir 

;:  -^  16  ft.  10'    9' 

[  24  ft.    0'    6' 


r 

53  ft.    8'    2''    Ans. 


DUODECIMALS.  307 

2.  Add  12  ft.  9'  IV  A"\  23  ft.  1"  10'",  and  10'  6"  9'". 

3.  From  21  ft.  T  10"  take  15  ft.  9'  4". 

Process-  /  ^1  ft.    7^10^' 

PROCESS.    ^i5ft^     9/     4// 


5  ft.  10^    %''     Am. 


4.  From  the  sum  of  30  ft.  8"  4'"  and  14  ft.  T  10'",  take 
their  difference. 

466.  MULTIPLICATION  OF  DUODECIMALS. 

5.  Multiply  13  ft.  7'  8"  long  and  6  ft.  5'  wide? 

Process.  Multiply  first  by  b'  and  tlien  by 

6  ft.,  and  add  the  partial  products. 
^^  ^^'    Tft  r  Since  1XtV  =  tV,  TVXTV=Th, 

g-^— g/  '^77^>7  Ti:rXTV=T7W  etc.,  feet  X  primes 

81  ft'  1(K     0^^  (^^  twelfths)  must  produce  primes; 

o«  ...      1^7^  2^/  4///     4  primes  by  primes,  seconds;  seconds 

by  primes,  thirds;  and,  generally, 
the  denomination  of  the  product  of  any  two  denominations  is  de- 
noted by  the  sum  of  their  indices. 

6.  What  are  the  superficial  contents  of  a  board  9  ft.  7' 
4"  long  and  10'  6"  wide  ? 

7.  What  are  the  solid   contents  of  a  block  of  marble 
7  ft.  6'  long,  2  ft.  8'  wide,  and  1  ft.  4'  thick? 

Note. — The  answers  to  the  5th  and  6th  problems  are  in  square  feet 
and  duodecimal  parts  of  a  square  foot,  and  the  answer  to  the  7th 
problem  is  in  cubic  feet  and  duodecimal  parts  of  a  cubic  foot  (Art.  464). 


467.  DIVISION  OF  DUODECIMALS. 

8.  Divide  87  ft.  6'  2"   4'"  by  13  ft.  T  8". 

Process.  The  process  is  the  reverse  of 

Dividend.            Divisor.  that  in  multiplication.    For  con- 

87  ft.     6^  2^^  4''''^  )  13  ft.  7^  8''^  venience  in  multiplying,  place  the 

81  ft.  10^                     6  ft.  5^   Q'nt.  divisor  at  the  right  of  the  divi- 

5  ft.     8''  2'^  4^^"^  dend,  and  the  terms  of  the  quo- 

5  ft.     8^  2''  4'''  tient  below  those  of  the  divisor. 


308  COMPLETE  ARITHMETIC. 

9.  Divide  62  ft.  ir'  W"  by  8  ft.  6'  %". 
10.  Multiply  10  ft.    5'   8"   by  3   ft.   10',  and  divide  the 
product  by  5  ft.  2'  10". 

PERMUTATIONS, 

468.  Permutations  are  the  changes  of  order,  which  a 
number  of  objects  may  undergo,  and  each  object  enter 
once  and  but  once  in  each  result. 

(cha 

469.  The  diagram  at  the  right  shows  f  ^«  1  ^^«  • 
the  number  of  permutations  of  1,  2,  and       al        ^     , 

3  letters.  [  ah'l  ach 

(  ahc 

The  letter  a  permits  no  change  of  order.  The  letter  h 
may  be  placed  before  and  after  the  letter  a,  giving  two  (1X2) 
permutations  of  two  letters — ha,  ab.  The  letter  c  may  be 
placed  before,  between,  and  after  the  two  letters  ab;  and  the 
same  for  b  a,  giving  six  (1X2X3)  permutations  of  three 
letters. 

A  fourth  letter,  as  d,  may  evidently  occupy  four  different 
positions  in  each  of  the  six  combinations  of  these  letters, 
giving  twerity-four  (1x2x3X4)  permutations  of  four  letters. 

In  like  manner  it  may  be  shown  that  the  number  of  'permu- 
tations of  any  number  of  objects  is  equal  to  the  continued  product 
of  all  the  integers  from  1  to  the  given  number  of  objects  inclusive. 

PKOBLEMS. 

1.  In  how  many  different  orders  may  6  boys  sit  on  a 
bench  ? 

2.  In  how  many  different  orders  may  all  the  letters  in 
the  word  permutation  be  written  ? 

3.  How  many  permutations  may  be  made  of  the  nine 
digits  ? 

4.  How  many  different  combinations  of  eight  notes  each 
may  be  made  of  the  octave? 


RULES  OF  MENSURATION.  309 


ANNUITIES. 

470.  An  Annuity  is  a  sum  of  money,  payable  annu- 
ally, for  a  given  number  of  years,  for  life,  or  forever. 
The  term  is  also  applied  to  sums  of  money  payable  at  any 
regular  intervals  of  time. 

471.  A  Certam  Annuity  is  an  annuity  that  is  payable  for 
a  given  number  of  years. 

A  Contingent  Annuity  is  an  annuity  payable  for  an  uncer- 
tain period,  as  during  the  life  of  a  person. 

A  Perpetual  Annuity  is  one  that  continues  forever. 

472.  An  Immediate  Annuity  is  an  annuity  whose  payment 
begins  at  once. 

A  Deferred  Annuity  is  an  annuity  whose  payment  begins 
at  a  future  time. 

473.  The  Forborne  or  Final  Value  of  an  annuity  is  the 
sum  of  the  compound  amounts  of  all  its  payments,  from  the 
time  each  is  due  to  the  end  of  the  annuity. 

The  Present  Value  of  an  annuity  is  the  present  worth  of 
the  forborne  or  final  value. 

Note. — The  principal  applications  of  the  subject  of  annuities  are 
in  leases,  life  estates,  rents,  dowers,  life  insurance,  etc. ;  and  the  prob- 
lems arising  are  readily  solved  by  means  of  tables  which  give  the 
present  and  final  values  of  $1  at  the  usual  rates  of  interest.  A  full 
discussion  of  the  principles  involved  in  the  construction  of  these  tables, 
can  not  well  be  presented  in  a  school  arithmetic. 


KULES  OF  MENSURATION. 
474.  Surfaces  and  Lines, 

1.  To  find  the  area  of  a  rectangle,  Midtiply  the  length  by 
the  width. 

2.  To  find  either  side  of  a  rectangle,  Divide  the  area  by 
the  oilier  side. 


310  COMPLETE  ARITHMETIC. 

3.  To  find  the  area  of  a  triangle,  Multiply  the  base  By  one 
half  of  iJie  altitude. 

4.  To  find  the  area  of  any  quadrilateral  having  two  sides 
parallel,  Multiply  one  half  of  the  sum  of  the  two  parallel  sides 
by  the  perpendicular  distance  between  them. 

5.  To  find  the  circumference  of  a  circle, 

1.  Multiply  the  diameter  by  3.1416.     Or, 

2.  Divide  the  area  by  one  fourth  of  the  diameter. 

6.  To  find  the  area  of  a  circle, 

1.  Multiply  the  square  of  the  diameter  by  .7854.     Or, 

2.  Multiply  the  square  of  the  radius  by  3.1416.     Or, 

3.  Multiply  the  circumference  by  one  half  of  the  radius. 

7.  To  find  the  diameter  of  a  circle,  whose  area  is  given, 
Divide  the  area  by  .7854,  and  extract  the  square  root  of  the  quo- 
tient. 

8.  To  find  the  side  of  the  largest  square  that  can  be  in- 
scribed in  a  circle,  Multiply  the  radius  by  the  square  root  of  2. 

9.  To  find  the  side  of  the  largest  equilateral  triangle 
that  can  be  inscribed  in  a  square,  Midtiply  the  radius  by  iJie 
square  root  of  3. 

10.  To  find  the  area  of  an  ellipse,  the  two  diameters  be- 
ing given.  Multiply  the  product  of  the  two  diameters  by  .7854. 

11.  To  find  the  surface  of  a  sphere, 

1.  Multiply  the  circumference  by  the  diameter.    Or, 

2.  Multiply  the  square  of  the  diameter  by  3.1416. 

12.  To  find  the  entire  surface  of  a  right  prism  or  right 
cylinder.  Multiply  the  perimeter  or  circumference  of  the  base  by 
the  height,  and,  to  the  product,  add  the  surface  of  the  tivo  bases. 

13.  To  find  the  convex  surface  of  a  pyramid  or  cone, 
Midtiply  the  perimeter  or  circumference  of  the  base  by  one  half 
the,  slant  height. 

14.  To  find   the  hypotenuse  of  a  right-angled   triangle, 


HENKLES  METHOD  OF  WRITING  DECIMALS.         311 

Extract  the  square  root  of  the  sum  of  the  squares  of  the  other 
two  sides. 

15.  To  find  the  base  or  the  perpendicular  of  a  right- 
angled  triangle,  Extract  the  square  root  of  the  difference  be- 
tween the  square  of  the  hypotenuse  and  the  square  of  the  other 


476.  Contents  of  Solids. 

1.  To  find  the  solid  contents  of  a  rectangular  solid,  Mul- 
tiply the  length,  width,  and  thickness  together. 

2.  To  find  either  dimension  of  a  rectangular  solid,  Divide 
the  solid  contents  by  the  product  of  the  other  two  dimensions. 

3.  To  find  the  solid  contents  of  a  cylinder,  Multiply  the 
area  of  the  base  by  the  altitude. 

4.  To  find  the  solid  contents  of  a  sphere, 

1.  Multiply  the  cube  of  the  diameter  by  .5236.     Or, 

2.  Multiply  the  surface  by  one  third  of  the  radius. 

5.  To  find  the  solid  contents  of  a  cone  or  pyramid, 
Multiply  the  area  of  the  base  by  one  third  of  the  altitude.  ' 

6.  To  find  the  solid  contents  of  the  frustum  of  a  cone 
or  pyramid,  To  the  sum  of  the  areas  of  the  two  bases,  add  the 
square  root  of  their  product,  and  multiply  the  result  by  one  third 
of  the  altitude. 

7.  To  measure  timber,  as  planks,  joists,  etc.,  by  board 
measure,  Eind  the  number  of  square  feet  in  one  surface,  and 
multiply  the  residt  by  the  thickness  in  inches. 

HENKLE'S  METHOD  OF  WRITING  DECIMALS. 

476.  It  is  seen  from  the  decimal  scale,  that  the  teris  of 
any  number  of  tenths,  the  hundreds  of  any  number  of  hun- 
dredths, the  thousands  of  any  number  of  thousandths,  etc., 
each  falls  in  the  order  of  units  when  the  decimal  is  expressed 
decimally.      Thus,   42  tenths,  written  decimally,  is  4.2,  the 


312      .  COMPLETE  ARITHMETIC. 

4  (tens)  falling  in  units'  order;  1265  huridredths,  written 
decimally,  is  12.65,  the  2  (hundreds)  falling  in  units'  order; 
and  425  thousandths,  written  decimally,  is  .425  or  0.425,  the 
0  (thousands)  falling  in  units'  order.     Hence,  the  following 

Rule. — To  write  a  decimal,  Begin  at  tJie  left  and  write  the 
term  corresponding  to  the  name  of  the  decimal,  in  the  order  of 
units. 


SCHUYLER'S  CONTRACTED   METHOD   OF  MUL- 
TIPLYING DECIMALS. 

477.  Rule. —  Write  the  midtiplier  so  that  its  units'  place  shall 
fall  under  that  term  of  the  multiplicand  whose  order  is  one  lower 
than  the  lowest  required  in  the  product.  Multiply  first  by  the 
units,  if  any,  carrying  to  the  nearest  unit  from  the  part  rejected. 
In  7nultiplying  by  a  term  on  either  side  of  the  unit,  commence 
with  that  term  of  the  multiplicand  as  far  on 
Process.  ^j^^  ^^^^^  ^^^^  ^j  ^j^^  ^^^^  ^j  ^^^  midtipli- 

12  S45  ^^^^  ^^^^  ^'^^  units^  place  of  the  multiplier, 

^^  carrying  to  the  nearest  unit.      Add  the 

34564  partial  products,  reject  ike  sum  of  the  last 

jgg  column,  after  carrying  to  the  nearest  unit. 

_     17  Point  off  the  required  number  of  decimal 

42.669  places  in  the  sum. 

Note. — Before  commencing  to  multiply  by  any  term  of  the  multi- 
plier, it  is  convenient  to  mark  that  term,  also  the  term  with  which 
we  commence  to  multiply  in  the  multiplicand. 


THE 

Eclectic  Series  of  Geographies, 

BY  A.  VON  STEIN WEHR. 

The  JPrhnart/  Geofjraphy,  No,  1.  The  plan  of  this  book  is 
natural,  the  language  simple,  and  the  definitions  and  descriptions  exact. 
Illustrated,  small  4to. 

The  Tntei'tnediate  Geogrcvphy,  No,  2 ;  for  more  advanced 
classes.  Contains  the  leading  principles  of  the  science,  so  arranged  as 
to  give  correct  ideas  to  pupils  without  requiring  the  constant  aid  of 
the  teacher.     Full  instructions  in  Map  Drawing.     Illustrated,  large  4to. 

The  School  Geography ,  No,  S ;  embraces  a  full  Mathemat- 
ical, Physical,  and  Political  description  of  the  Earth,  and  is  intended 
for  the  highest  classes  in  this  branch  of  study.     Illustrated,  large  4to. 


GRADED  SCHOOL  ARITHMETICS: 

A  New  Series,  on  an  Entirely  JVew  JPlan, 
BY  E.  E.  WHITE,  M.  A. 

Whitens  JPriniary  Arithnietic,  The  distinguishing  feature 
of  this  book,  as  well  as  of  the  series  of  which  it  forms  a  part,  is  the 
complete  union  of  Mental  Coral)  artd  Written  Arithmetic.  This  is  secured 
by  making  every  oral  exercise  preparatory  to  a  written  one,  and  by 
uniting  both  as  the  essential  complements  of  each  other.  Illustrated, 
16mo.,  144  pages. 

Whitens  Intermediate  Arithmetic,  This  treatise  possesses 
three  very  important  characteristics :  1.  It  is  specially  adapted  to  the 
grade  of  pupils  for  which  it  is  designed,  and  is  not  an  abridgment  of 
the  Complete  Arithmetic.  2.  It  combines  Mental  and  Written  Arith- 
metic in  a  practical  and  philosophical  manner.  3.  It  faithfully  embodies 
the  Inductive  Method.    Illustrated,  16mo.,  192  pages. 

Whitens  Complete  Arithmetic,  This,  like  the  preceding  books 
of  the  series,  combines  Mental  and  Written  Arithmetic.  It  embraces 
a  particularly  full  and  clear  presentation  of  the  subject  of  Percentage, 
with  its  various  applications.  Abridged  Methods  of  operation  are  given 
under  most  of  the  rules.     Illustrated,  12mo.,  320  pages. 


EACH    SERIES    COMPLETE    IN    THREE    BOOKS. 


>  D    OOODil 


ECLECTIC    EDXJC^TIO]>f^L    SERIES. 


THE 

ELEMENTS  OF  NATURAL  PHILOSOPHY, 

By  SIDNEY  A.  NORTON,  A.  M. 

Norton's  Natural  Pliilosophfj  is  the  result  of  many  years' 
experience  in  teaching  the  science  of  Physics.  The  topics  are  consid- 
ered in  their  logical  order,  methodically  developed,  and  thoroughly 
illustrated  and  enforced. 

While  due  attention  has  been  given  to  the  recent  progress  in  Physics, 
including  the  latest  methods  and  inventions,  it  has  not  been  forgotten 

that  al^    ^*"^^''   <»»»ia   onnollir   -frAcVl    fn  flip  t.vrn 


N 
cardi 
studi 


EC 


a  ha 
been 
ence 
gene 


'  n 


)r  have 
'  pupils 


HIP. 


ds,  and 
em  has 

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THE 

j:  :  :  !'      SERIES 

■  GEOGrvAPIilES 

>X  STpiNWEHK  k  D.  G.  BRINTO 

The  series  cbnsistd  of  three  b'^oks:  a  Primary,  an  Inte 
and  a  School.  Geographiy.    Tiiough  each  buok  of  tb' 
from  the  omers'  in  scapei  and  treatment,  tKo  arrant 
att^jpiatmaVd  is  the  same  -a  all.    This  uniformit 
JpfntatesthG;  progress  of  the  pupil,  as  hc.is  not  '  : 
^ew  classifica.tion,  or  to  master  a  new  drrang^ 
be  passes  fron^  the  study  of  one  book. oif  the  '^  / 

The  teach/  r  is  furnished  with  every  mea:-    ' 
is  not  forceffj,  at  each  step,  to  classify  a  mass 
facts,  or  to  adjd  an  extended  commentary  o 

The  Pnll^a^^ry  Geogfra].^y  contains,  the 
-ciorice,  stated!  in  plai  j,  simple  la:iguage. 

Che  Injt.ern[iedi^^  oh    ^eography  is'  fuller 
sufh^ientf^J^  a.  r^^hm   ,  loui-se.     The  map-dra  ' 
]      after  uhe  "'escrii/'  ..  the  political  divisioi^ 

;^  ■.     prv  siinple  <    0,' and*  will  prove  an  inval 
>iA  the  form,  physical  features,  and  p 
t-n'tte  '  ' 

;:ool  Geography  is  designed  tcK  complete  the  course, 
anoucj  t(ipics  art^  niort  fully  treate^Jah  this  book  than  in  the    ai 
':itl\   and  it  contains  ft  complete  outline  of  niathematicii!   a 
^  ography.     The  sectional  maps  ;ire  .so  constructed 
■le  for  refeiorce. 

'   Ishers  invite   "".pecial  attention  to  the  Maps,  which    ' 
■\V'^"^\>'?' titles  arid  artistic  execiltion,  are  greatly  superior  to  those  ol  .  ■/ 
A    ^' <v^^  ^/^ries    of  school   geographies.     I'he   mountains,   plp,ii>'-,  ai/ 
Y\     •e,*"'^-\;t»    "  ^^'^  ^^  skillfully  treated  that  the  maps  re,seni^>f,.    '^ise, 
V      0*^  ^^  projection  is  the  pclyconic — that  on  whi<  b  >"-  ^-'uri..> 

V)aat  Survey  are  constructed.     T1j<  ^vot>d-cuj 

/,'        iuive  been -engraved  by  the  best  ar  "  *^-vP^ns( 

red  to  render  tlte  series  attractive, 
ct'o  series  is  i.iC  ^nly  one  publisher  '  }^^icl], 

iho  New  Geoffrai^hy— the  product  of  '-^^'ibory 

;r  and  Humb<iTdt — with   scientific   pre'  ^  );'^'''  ''JRd 

...ilicity  ;  aiid  it  i-  "f^i-n' T  i,^  fhe  educajbiorul  p.    ^^,.,,.    '""."'  ^*«- 
ief  that  it  wi"  be  j  <^  mostemluenV^'y     '^^  '^""es 

-chool  !'!.  «  -iiphie.'<  >  ''v/CTiT?  p 

WILSON,  ^■^^''- 


